Kind of depends on if you really want a house or if a flat would work for you.
For houses Totterdown & Bedminister – Windmill Hill probably out of your budget. Totterdown is a bit nicer than Bedminister IMO. Great for town and for Temple Meads, and around Victoria Park there are three really nice pubs. One thing with Totterdown is how close it is going to be to the new Arena – not sure if this will be good or bad for house prices.
North of the river you’re getting out towards Horfield which you might find is a bit far out for town and for getting to Bath.
If flats are alright then you’ve got more options north of the river, like Redland, Gloucester Road etc. Not sure on older houses always needing work – plenty of newish flats and houses that are pretty shoddy…
Bike friendly houses in Bristol are pretty difficult – garages feel vulnerable, as do sheds. We went for a garden with no external access, so thieves either have to go through the house or through lots of neighbours gardens.
Best of luck!