An average budget of these gifts for greeting new neighbors is between 1000 yen to 2000 yen. If you're a struggling student, 500 yen is acceptable.

Yes, tiny hand towels are popular, but now that people are becoming more picky about the designs they want to use, consumable gifts, in other words things that disappear, are more appreciated in cases like these where you don't really know much about the receiver.

Typical gifts are tea and sweets. Why not drop by a confectionary shop of your choice and ask for advice. A low-priced item in a pretty box would do.

Speaking from experience, it would be nice to put your name on the gift so that your neighbors can remember your name (as well as names of your family members if you're living with any). Traditionally, the Japanese use what we call "noshi" but a tiny gift card is often used as well.

Yes, those directly next to you and the people directly above and under your floor are fine. The idea is to greet people that might be affected while living there, so if you feel that there is anyone else that might be affected, you can give it to them as well. For example, we have people living across the street from our backyard fence, and it was good that I went to greet them, because I ended up growing plants that had leaves falling on their premises.

Just introduce yourself by telling them your name and that you just moved in (or moving in soon) and say "yoroshiku onegaishimasu". You can just say "kore, douzo." as you're handing them the gift. You don't really have to make a big proper deal out of it unless you really want to.

If no one is home, you can leave a note in their mail box, or some people just give up after trying a couple of times.