I was wondering why there is a を there. I thought that if the first verb is intransitive, and if the second one is transitive you could put a を?
Right. Here 土日 is the object of transitive 過ごす. Please refer to:
From @naruto's answers in the pages above:
the first verb before て describes how the second verb is done, in which situation the second verb is done, etc. That's why 重ねて尋ねる means "ask in a repetitive manner" rather than "repeat, and then ask".
[in a phrase みなさんのパフォーマンスを通して聞く] Note that the object of みなさんのパフォーマンス is 聞く, not 通す. 通して is the te-form of 通す, and here it's working like an adverb that describes how the main action (聞く) is done. Compare: 音楽を座って聞く ("to listen to the music while sitting"), リンゴを急いで食べる ("to eat an apple in a hurry").
As you can see, the first verb 寝る in Verbて+Verb 寝て過ごす can describe how the second verb 過ごす is performed. 寝て is working like an adverb that describes how the main action 過ごす is done. You can understand your example in the same way as @naruto's examples 音楽を座って聞く "to listen to the music while sitting" and リンゴを急いで食べる "to eat an apple in a hurry": 土日を寝て過ごす "spend Saturday and Sunday while sleeping".
What you're waiting for is to spend Saturdays and Sundays sleeping, and a lifestyle of a working adult who faces Monday mornings with hopelessness.
I'm afraid you're not parsing the sentence correctly. One grammar point you should be aware of is:
「V₁てはV₂」 = "do V₁ and V₂ repeatedly/habitually as in a pair" "habitually do V₂ right after V₁ as a paired action", in this part:
土日をただ寝て過ごしては、絶望の月曜の朝を迎えるだけ
which would mean "just spend Saturdays and Sundays sleeping, and face Monday mornings with hopelessness, repeatedly/habitually", "just repeat spending Saturdays and Sundays sleeping, and facing hopeless Monday mornings". Point is that the two actions 土日をただ寝て過ごす and 絶望の月曜の朝を迎える are just repeated habitually, in a pair.
For this grammar point, please see:
Now I believe you can see why 土日をただ寝て過ごしては、絶望の月曜朝を迎えるだけ should modify 社会人生活, as @aguijonazo pointed out in the comment. The structure of your sentence is:
待っているのは(V₁ては、V₂の)社会人生活ですよ。
As you have already noticed, this is a cleft sentence. This answer may also help. I think you can understand it as:
「社会人生活が(あなたを)待っている」⇒「(あなたを)待っているのは、社会人生活だ」
A life as a member of society awaits you ⇒ What awaits you is a life as a member of society
So, to summarize all the above, the sentence should be parsed as and literally means:
待っているのは(土日をただ寝て過ごしては、絶望の月曜の朝を迎えるだけの)社会人生活ですよ。
What awaits you is a working businessman's lifestyle (in which you just repeatedly and habitually spend Saturdays and Sundays only sleeping and face Monday mornings with hopelessness).
土日をただ寝て過ごしては、絶望の月曜朝を迎えるだけ
modifies社会人生活
.