As an example, the hip place to be right now is around Telliskivi that is constantly adding new cafes, shops etc. Depending on the size and condition of the apartment, you should expect to pay between 300-500€ per month plus 80-200€ (the latter in the coldest winter months if the place isn't well insulated) for heating, water and electricity. You can definitely get something for 250 if you're okay with being further away in a small and not that good looking apartment as well.
Expect to add at least 100€ to that if you want to be really close to downtown.
Depends on your standards really.
These two sites both have an English version and will have about 90% of all available apartments listed. You can do map based searches etc. This should give you a much better idea.
Also something a lot of people don't expect: whenever renting you'll usually have to pay the first months rent up front, a deposit also equalling the first months rent and a commission to the blood sucking real estate agent (yes really, the renter has to pay it). You can also find apartments directly from owners (no more than 10% of the listings) in which case it's only 2x the monthly rent.
As a final note, judging by your name and location (pardon me if I'm wrong) you're probably Arabic?
Estonia only really opened up to the rest of the world in roughly the last decade and as such there's still quite a few people who are majorly racist. They're definitely in the minority and most younger Estonians are super open minded but if you go apartment hunting there's a fair chance you'll run into some douchebags that might slam the door on you. There's an article about a few immigrants experiences along these lines. Might I suggest you ignore them and carry on. You wouldn't start arguing with a Trump supporter would you?
But don't worry, we're mostly really nice people. Just a heads up.