Fur Elise is not really for beginners, it is more of an intermediate level piece. My teacher's rule is that after a couple of weeks, I should be able to play a piece I have learnt (not concert level of course, but going through it without any major issues like notes or tempo). There might be exceptions for professionals working on difficult repertoire, but from an amateur point of view, I think it is more motivating this way and that it helps build confidence too. I know some teachers who would teach only one or a couple of difficult pieces in 1 year, but is it really worth it?
I think that if after two months you are still struggling, the piece is probably above your level. The first part is the easiest to play, maybe focus on playing the first part for now and keep the rest for when you are more advanced?
Technically speaking, if you do not want to slow down during the rather fast passages, you need to be comfortable playing a chromatic scale (let's say at a 120 tempo), an a minor arpeggio, and the same goes for the c major scale. I agree with Danielo about the patience and self discipline, although I have doubts about the fact that playing really slowly for two months will improve your speed. I think it will improve your accuracy only. You need to be consistent in your practice for months/years if you want to build a strong technique, and gradually speed up.