I don't know if this is still relevant. First, the name of the city is BUDAPEST. Second, I am both Canadian and Hungarian, and I am an alum of U of W. I visit
Hungary once or twice a year and stay for a couple of months each time.
Budapest is safe, has very good public transportation, which is how I get around there most of the time. I find that a monthly or bi-monthly pass works out best. I think living there is pretty expensive even for me. A pair of nice shoes cost over 100 US$. If you have a well paying job and a free apartment, you might do well. The postal code for your address is a good area. In fact, I would say that most any area is safe. I walk around alone at night in Bp. which is something I wouldn't do in LA. Some areas are more expensive and desirable, but not more safe. I think language will not be a big problem, especially with younger people. The language schools do a brisk business in teaching English. Of course, there are a few older people (such as myself) who speak English very well.
I will be going to Hungary at the end of May this year. If I can help you any way and if your time and mine coincide, you can respond to this comment and I will give you my Hungarian cell number and try to help you with any questions you have. Good luck!