Why does 前 mean "past" in terms of time, but "forward" in terms of direction?
So far I was able to find that 前 is constructed from Chinese: 舟 and 止, which can be interpreted as "leaving a footprint", hence the "past" meaning, but it seems kind of far-fetched.
There are many issues with the question, because it doesn't differentiate between words and characters. Fully addressing the question requires a discussion of the following:
Firstly, there is no demonstrable link between the character「前」and the meaning forward. The oracle bone version of「前」looked like
That is, a compound of「止」(foot) and「凡」(a tray, now written「盤」). The original meaning was to wash one's feet (e.g. in a basin), and sometimes to emphasise the meaning, drops of water「氵」were added around the foot, and「凡」was replaced with「用」(a tub/bucket, now written「桶」).
The original meaning was extended to mean wash in general, now written「湔」.「凡」is often confused with「舟」(seen in many characters such as「盤」, where the tray-shape is morphed into「舟」), because「凡」and「舟」looked very similar in many bronze inscriptions:
(Bronze script forms; Left: 舟, Right: 凡)
This leads to the form「歬」, and by this time「凡」and「用」were long gone from the character and「歬」was used for the meaning forward.
The character didn't stop changing, however, and eventually「刀」was added to represent the meaning cut, later written「剪」(to emphasise the meaning cut differentiated from forward via the addition of yet another「刀」). An overview of the change over the years looked something like this:
Hopefully this demonstrates the importance in separating the concepts of words and characters; the one series of characters「前」has been used to represent three completely different words (meaning wash, forward, cut) which merely sounded similar.
Now that we've determined that the character has nothing to do with the meaning, the second point of interest is the association of forward and past, and backward and future. Neurobiology/psycholinguistics terms the description of temporal directionality using spatial language as spatiotemporal metaphors, noting that words that describe spatial locations are borrowed in most languages around the world to also describe temporal sequencing. Studies relating to the English language typically categorises these as time-based and ego-based metaphors:
It is clear that English also has a metaphor equating front as past, which is directly translatable to「前」; however it is the time-based before rather than the ego-based forward.
Japanese「まえ」(etymology/historical spelling:「[目方]{まへ}」, the direction which the eyes are looking in), as a spatial term is an easy enough correspondence to Chinese /*[dz]ˤen/ (front), but did it carry a connotation of before, past prior to the importation of kanji? The presence of a few other Japanese vocabulary items specifying the past suggests a preference towards ego-based metaphors:
To summarise:
Probably for the same reason that someone who is "ahead" of someone else is both in front of them physically and also arrives earlier. For that matter, the English word "before" also shares the same meanings of "earlier" and "in front" (though the latter meaning is rarer in current usage).
Equating temporal directions with physical ones is a tricky matter that's more complicated than it might intuitively seem. 前 is actually the simpler Japanese example here, because it is at least very consistent in which direction it refers to in each context. 先 is so nebulous that it can actually refer to the past or future.
As a Chinese native speaker who knows a little bit about Japanese, I would like to talk about this character on a Chinese perspective. I believe the origin of your question can be traced to the Chinese origin.
Picture source: http://www.fantizi5.com/ziyuan/ and then search "前".
Translation: Original character: (止 over 舟). This character is from 舟 and 止, where 舟 is a boat, and 止 is a foot, indicating the boat is going forward.........
In Chinese, 前 has almost the exactly same meaning as in Japanese - the first is something in front of something. E.g.: 面前, which literally means "in front of the face", or just "in front of (a person)"; 站前, which is used by many cities as "the area in front of the (main) train station". This can be considered as a priority queue (sorry, but I'm a computer scientist, and this is a computer science term, but I do think it is not hard to understand), 前 means A has a higher priority than B.
The second meaning is about time - which means "before". However, in Chinese, 前日 means "the day before something happens" (e.g.:结婚前日, "the day before wedding"). Interestingly, 前天 means "the day before yesterday", though people usually treat 日 and 天 as two characters with the same meaning. More interestingly, "先前几天/先前几日" in Chinese both have the same meaning: "preceded days before today (or someday)", "昨天/昨日" both mean "yesterday". Nevertheless, 前 means "preceding time". Time can also be considered as a queue - the preceded time can be considered as having a higher priority. Or front another perspective, the preceded time is "in front of" the following time.
Another meaning is "proceed". For example: "前进" literally means "proceed forward", which means "proceed". When proceeding, it means to get higher priority.
All in all, 前 means "a higher priority" as a state/time/etc, or "to get a higher priority". Therefore, it means going forward.
I saw someone said that he/she is "always confusing the meanings of 先日 and 前日" in Japanese - it's normal, as you can see from above, they are complicated from the root (which is Chinese itself) of this question.
P.S., for Chinese, the words with "天" that can be changed to "日" without changing the meaning (e.g. "昨天/昨日", "先前几天/先前几日", "三天/三日 (three days)", "星期天/星期日 or 周天/周日(Sunday, 周天 is rare, 星期天 and 周日 are most used, 星期日 is a little bit formal)", "工作天/工作日(Weekdays, though the first one is rare)"), the one with "天" is more informal, and the one with "日" is more formal. Even for most words with "日" that is not interpreted as "day" (e.g.: "日" itself can either mean "sun" or "day" (and sometimes can be used as f-word in Chinese), but "天" only means "day"), it's more formal than its synonyms (e.g.: 日 vs 太阳, both mean "the Sun", and the first one is more formal; same when interpreting "日" as a swear word, it is more formal than its synonyms).
It's hard to feel the inner meaning of a word when learning another language. I personally learn English (since 2001-2002, when I was 4-5) and German (since 2014), and I do speak better English than most of my Chinese friends - but still worse than real native speakers...... It is necessary to be soaked in that environment in order to be a better user of one language.
My understanding is that it's analogous to English usage of the word before.
In some cases, before in English has the connotation of something that occurs prior:
学校から帰る前に、図書館に行かなきゃ。
Before I go home from school, I have to go to the library.
In other cases, before has the meaning of something that is in front of you (i.e. standing before you):
急にアリスさんは僕の前に立ち上がって、「好きだよ、きみのこと」と言った。
Suddenly, Alice stood up before (or, in front of) me, and said "I love you".
Because one thing happens ‘before’ another thing in time. Actually it has the meaning of ‘in front of’ or ‘before’. So something can happen in front of you. Or something happens before another thing happens.
