Here are three different scenarios. What is the Hungarian for each?
1. I climb the mountain [I go all the way to the top]
2. I climb on the mountain [I move up the mountain but may or may not go to the top]
3. I climb to the mountain [I climb some lower mountains in order to reach the base of the main mountain]
It is like "until"
Elmegyek a hegyig. I go to the base of the mountain (and possibly stop there and don't climb it)
It is often used with time:
Megcsinálom ezt a munkát négyig. I will do this work until four a clock.
I have to follow this conversation haha. I don't think I am completely ready for some of this - it's advanced at the moment, but I did see the -ig suffix meaning until, and I have seen that when trying to read YouTube comments. VVsey, Marton, and Zsuzsi mentioned meg- and el- (not just away as I have seen, but completion?) prefix.
It sounds odd to me, too, in both language versions ;-)
In the hungarian sentence it looks like a mixup of different grammatical rules. In combination with verbs like "akar", "lehet", "tud", "kell" and a few others sentences appear like this:
Arra a hegyre én fel akarok mászni. / Én fel akarok mászni arra a hegyre. = I want to climb up that montain.
But as you said: negating a verb requires placing the prefix behind the verb.
So: Arra a hegyre én nem mászom fel!
Beside the case, you want to negotiate the meaning of the prefix: not climbing UP but DOWN. But then it would be "...nem felmászom."
OK, this is a special case here, let's say an emotional way of expressing what one is not willing to do, not in a million years. You were told about a nice easy walk and suddenly see this monster of a mountain in front of you.
So, this is a very emphasized, frequently emotional way of saying something.
There are many uses for this structure in other situations, as well. Preverb - something in-between - verb. That something can be "is", "sem", "nem", etc.