Okay. Check out the following website, it has the map of Vancouver city only (
not any of the suburb cities) showing all of the Vancouver City neighbourhoods and there are itemized listings of what each neighbourhood is like.
Vancouver Neighbourhoods
The neighbourhoods that are closer to the downtown business core are the ones that you could reasonably expect to fall within the travelling time frame you're hoping for so if you're really lucky you might find something there within your budget.
Don't be getting your hopes up high with finding something in the more westerly neighbourhoods that are closest to UBC since they are wealthy neighbourhoods and anything that might be for rent there will likely be far and away above and beyond your budget for a residence.
The neighbourhoods further out from the downtown core and closer to the Fraser River would likely take longer than half an hour on bus transit during peak rush hour times unless you're in a neighbourhood that has rapid transit train going through it. See train map:
Interative Metro Map of Vancouver - JohoMaps
But those neighbourhoods that are further away to the south and east from the downtown core and don't have train going through them are the ones that happen to have the most potential for you finding a residence that is within your budget. So take into consideration that you may have no choice but to sacrifice the convenience of time saved in favour of money saved for a comfortable dwelling.
Regarding the Downtown Eastside neighbourhood (DTES) or Chinatown neighbourhood - be wary of them. Yes, they are very close to the downtown business core but they also happen to be Vancouver's skid row and homeless district and are dangerous, so read about them to find out why you should avoid them and then forget about them.
Good luck, I hope you will find something suitable/affordable in Vancouver. If you can't find a suitable residence within your budget in Vancouver City you will have to consider the surrounding suburb cities, so I'd suggest then that you look at areas that have rapid transit trains going through them.
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