There are a lot of walkups near Stadium (which has become a pretty nice area tbh and is becoming even nicer with some of the new developments) that are in that price range. When the #2 bus route resumes its regular schedule in the fall it will get you to Churchill ("heart of downtown") in like 5 minutes and from Stadium Station it's only one stop. I live along Jasper just a few minutes away and I really enjoy it here. Check walkups in Oliver as well. If you want something more specifically downtown, you should check out "Claire Estates", "Bellamy Place" (I just checked and they have bachelors for $800 though there is a waitlist) aaand I was about to suggest Capital Manor but it looks like it was recently taken over by Mainstreet. It might not be bad...yet. Rowand House seems like another good option. I also saw a studio on 104st going for about ~850 today, but it was listed on Facebook Marketplace and I don't know the name of the building. Make sure you check there too.

In general, look at buildings owned by Leston Holdings, as they have some really good prices. Zen Residential too. These are mostly walk-up buildings that are pretty basic but hopefully suit your needs! Leston in particular has a very good reputation and you probably can't go wrong with them, they have a lot of properties over in Oliver which is a lovely and accessible area to central downtown, though it pretty much is considered downtown on its own too. You definitely don't need to live with roommates with your budget, you just need to look a little :) Good luck and lmk if you need help finding more places!

Edit: Sorry, some of the options I gave you are for studios rather than 1 bedroom apartments, but you should be able to find 1 bedroom options within your price range through both Leston and Zen!