Hi there! A local here
Ruzafa sure is a great neighborhood lo live but definately more pricey than the rest of Valencia. Valencia is really walkable and cyclable and everything is pretty much within a 15-20 minute reach so maybe looking for a more economic location doesn´t make a big difference. There is not really a part in city center you should avoid so I find it hard to make a mistake there. Personal tip: The beach area (Malavrrosa and Patacona) although beautiful can be kind of empty and isolated during the week and specially in winter. I wouldn’t move there and rather rent a bike to go to the beach whenever you want. Just sayin’
For your reference an OK flat of around 80 sq meters with furniture should cost you 600 euros/month plus extra costs (water, electricity, etc) if you were to rent long term so for a three month stay I wouldn’t pay more the 700-800 for the whole month. Avoid touristy areas (Ruzafa, Carmen, Centro) if you don’t find anyone willing to rent for a couple of months.
Valencia is a fantastic city to spend a couple of months, don’t miss your chance!