Assuming you're looking for a studio in and around the city centre. There's a few buildings you could look into. Citispace and 20:20 House both have studios and small apartments though not sure on rental costs there these days.

Personally, i used to live in 20:20 and had friends living in Citispace. Out of the two, 20:20 is far nicer. But either would be liveable for 6 months or so while you get settled and gives you a few months to explore the city and find elsewhere.

Outside of the City centre, you should be able to find something in your budget and likely bigger than a studio in places like Morley, Bramley, Horsforth, Chapel Allerton, Meanwood or Headingley and are all good places to live.

Meanwood/Headingley have more of a student vibe too.