The verb forms used translated as "go" and "betrays" are the middle voice where a person acts on, by, or for themselves. So Jesus goes by himself and gives himself over. Jesus gives himself over "by the man" or "through the man". Judas is used as a means to an end. This verse makes it clear that Jesus is choosing these things for himself.
The verb translated as "is determined" and "has been decreed" is used by Jesus only here. It primarily means to "divide", but from that comes to mean "ordained" and "determined". It is in the form of a passive participle but it is introduced by a neuter article so the sense is that it has the sense of being decided by fate more than a person.