OK, for your questions:
Is it noisy? – Yes. That is the answer in I would guess 100% of the apartments, especially in the frum neighborhoods. Just based on the sheer number of people living in each apartment, the number of apartments in each building, and the number of buildings close to each other, you are talking lots and lots of people living close together, so things get noisy. If you want to go somewhere quiet, go to a desert island.
Frum neighborhoods – for example Ezrat Torah, Sorotzkin, HaMem Gimmel, Zichron Yaakov, Panim Meirot, Belz, Geulah, Mattersdorf, Unsdorf, Maalot Dafna, Sanhedria/Murhevet, Ramat Eshkol, Ramat Shlomo, Ramot (not Beit, but Gimmel and Daled are mostly frum and Aleph is half/half), Neve Yaakov.
Shuls – There is a shul on most streets in Yerushalayim, so that is not a big question unless you are physically handicapped and cannot walk a block or two.
Shopping – There are supermarkets in many main areas, but if you need to get a loaf of bread/milk and even a few extra things (higher prices), there is usually a makolet central in the neighborhoods. Again, unless you are physically handicapped, you can easily walk the distance.
Buses are the main source of transportation, unless you plan on taking taxis everywhere. Try to get ahold of Egged bus route maps so that you will know how to get where you want.