You're going to find it very difficult to find a place to rent for only 4-5 months. Almost all Chinese rental contracts are 1-year minimum. 

M you might be better sharing a place. Problem is finding somewhere. If you can read Chinese then check www.58.cn or www.ganji.cn

I lived in Tianjin for 6 years. Most places are starting to adopt the 6 month contracts instead of the 3 month ones. TEDA can be a little more expensive than the downtown area but hooker infested industrial centres (that were recently blown up with chemicals) are obviously in high demand. It really depends on how big a place you are looking for. I would hazard a guess at 3000+ a month. Please remember that you will have to pay the full contract amount up front + 1 months deposit. If you are going through an agent then you will also have to pay for that, downtown it is half a month, Jinghai it is a full month, I am not sure about TEDA. Good luck and remember to shop around, there are plenty of gems and TEDA is quite nice but you need a car to appreciate it.