"작년 같이 예비1번이 떠서 맘고생 하는것도 싫고"
떠서 (base form 뜨다) originally means to take off, fly or float up in the air (비행기가 떴다, 연(kite)이 떠 올랐다).
This sense is extended as slang for something suddenly becoming popular, or something appearing or being offered prominently for people to see, like a flashier version of "come up".
- 두 번쨰 음반이 떴다 = The second record hit it big.
- XX 가 여의도에 떴대 = They say XX was spotted in 여의도 (XX being some famous, sought after act)
- 화면에 모르는 사람의 메시지가 떴다 = An unknown guy's message appeared on the screen.
Your sentence probably means the person received a number one position (예비 1번) on the admissions waitlist, indicating their exam score was just below the cutoff and they would be admitted if anyone among those who passed the cutoff failed to register.
It is similar in meaning to 예비 1번을 받고 or 예비 1번이 돼서, 예비 1번을 받는 상황이 생겨서, etc. but 떠서 works well as it implies the news of it came up on its own (i.e. announced by the school independently) after a while rather than immediately.
As for 올해를 걸고 있습니다, it's more like he's counting(like betting) on this year (or betting this year on something depending on the rest of the sentence), as 걸다 here means betting or putting most of one's resources on the outcome of something.
X-를 Y-에 걸다 is the typical structure ("bet X on Y"), as in 올해 입시에 모든 것을 걸고 있습니다, "I am putting everything on this year's entrance exam".