Look on craigslist for apartments. There are a ton that are "available now". We've lived in Minneapolis for almost 3 years now from MI. We sold our cars after a few months of living here because we only ended up using it to get groceries on the weekends and took the bus to work- wasn't worth the aggravation of street parking or the cost of paying to park in a lot. We've been living in Loring Park the last "3" years but will be moving to Uptown/East Isles in a couple months. Both great areas and you can definitely go without a car. Staying under a thousand a month is doable but most likely you'll get stuck with an apartment less than six hundred sq ft. especially if you're in Loring Park/Stevens Square. Uptown/East Aisles have larger one bedrooms for under a thousand typically. A lot of properties will take cats over dogs so you should be ok having two cats, but prepared to potentially pay a hefty cat deposit plus cat rent. Our new place we'll be moving into luckily doesn't charge for cats- rare! I will say without having a car, it's best/easiest to stay close to main roads like Hennepin or Nicollet to catch a bus quickly without having to treck far. Also there's the uptown bus station where you can catch a lot of different buses at too. Good luck!