From Google Maps, you can see the location of Hampton is a bit beyond the north-south center, towards the "north" end of the city, and Batumi is not that large ie, can be walked.

practical/majority of cafe's/etc, end to end along sea-side approx 3km, or
diagonal from Hampton to CarreFour or AgroHub (regional traditional foods,bio-foods, etc), approx 2.5km;

many small, narrow streets towards northern end with many small shops/cafes. As go south and inland, streets are wider, blocks longer, but still most things, conservative-estimate, within 3km. There are buses that go up & down (north-south) street with many international brand hotels (Hiltons and Marriotts, one Radisson; no IHG's and Hyatt Centric still under construction and who knows, it might get reflagged before it opens)