The sentence is a popular saying that compares two ways of staying young. A face-lift is a cosmetic surgery that tightens the skin to make a person look younger on the outside. A mind-lift is a play on that term, meaning a change in attitude or outlook that keeps a person feeling young on the inside. The saying claims that changing your mindset works better than changing your face.
Only option 1 fills both blanks so the sentence makes sense and keeps the pun intact. The word "young" in the first blank sets up exactly what a face-lift is meant to achieve, and "face-lift" in the second blank is the direct match for "mind-lift", since both are lifts, just of different things.
Option 2 gives "at home ... egg", which has no logical or idiomatic connection to a mind-lift or a face-lift. Option 3 gives "light ... elevator", which uses the word "lift" in its literal sense of an elevator, but this breaks the pun the sentence is built on. Option 4 gives "away ... sleep", which also does not fit the meaning of staying young through a change in attitude. None of these three options preserve the wordplay between mind-lift and face-lift, so they are all wrong.