This answer extends on a lot of what @Earthliŋ♦ and @T.Allred have already posted.
The presence of one noun and one verb, in the absence of explicit particles, may imply that the noun is the subject and the verb belongs to that subject. Or, it may imply that the noun is the object of the verb. Given that, it comes down to which answer makes sense logically.
This may imply
Logically, however, this doesn't make a lot of sense because it means, "Oh, the bus, it has to go".
Or, it may imply
This doesn't make sense, either. It might mean "Oh, we have to walk down the bus".
As @T.Allred explained, 「あ、バスだ。行かないと。」would make this answer correct.
For the second choice,
Here, one noun, one verb, so again it's implied that the noun can be the subject or object of the verb. Of the two cases, it makes logical sense for the noun to be the subject, so this option is correct. This means, "Aw man, the bus left".
Response to additional question
Also, concerning, 行っちゃった, is it common to use 行く when describing the
motion of a vehicle? Wouldn't you rather use 出かける or 出発する?
Yes, it is common to use 行く when expressing that a car, train, etc., has gone, especially when you don't want it to have already gone （行ってしまいました/行っちゃった）.
The subject of 出かける is normally a person/people, and, per Earthliŋ♦, it means "to go out". So, although バスが出かける may be grammatically correct, from a logical standpoint it might be a little funny.
出発する literally means "to depart". Usage of 出発する, however, is common from a vehicle operator (driver/pilot) to passengers in the vehicle. It's also common when talking about vehicles with respect to its schedule. In the case of the question, 「あ、バス、出発しちゃった」isn't incorrect. However, あ、○○ ~ちゃった implies that the speaker is talking/thinking to him/herself, or the speaker is talking to a close friend in a colloquial manner. In such a colloquial situation, as @Earthliŋ♦ indicated, 出発する doesn't seem natural, much like, "Aw man, the bus departed" doesn't seem natural, either.