Two new discoveries here:

When goin up the stairs Snake won't plant the C4, but when going down he will.

If you press the crawl button right after exploding the C4 it will cancel the punch in the air and save some time. Special thanks to koji123456 for telling me that.

I tried 2 different strats here and this was the fastest I could get. There must be a faster way, though, as the record is a lot faster than this (3,85).