I used to visit a lot between mid 90s and early 00s. Changed out of all recognition - when I first visited it was dumplings and potato soup. It seems to now have a Brasilian restaurant.
I eventually used to stay here:
Carlton Bratislava
.... amongst other things, it was the German HQ in WW II. Around the corner is the Devin Hotel - smaller with a terrace on the river.
The reason to stay here or closer is that most places you want to see and visit to eat/drink are walkable from here. In front of here is a pedestrianised quarter of the city, probably 20 mins or so to walk across. It is all little lanes, small restaurants and bars.
Slovak Pub
Memory tells me this is close by - famous in the past for having the longest bar in Bratislava. I can see now much more upmarket.
If you walk through the old town to the Cathedral (in the communist era they built a motorway across the entrance), go through the underpass and you can walk up to the Hrad (castle). Again all little lanes and restaurants etc. Great views across the Danube, but the castle is a bit dull. Bratislava was Pressburg, a much less significant city than Vienna or Budapest in the Austro-Hungarian days.
This whole old town was the Jewish area - Robert Maxwell was born in Bratislava.
This is worth a visit if you're interested in WWII history and the Soviet influence on Slovakian history.
Slavín Bratislava
in a posh part of Bratislava. Dubceck spent his internal exile years here. I think his old villa may be open to view.
Hrad Devin
a short taxi ride away. Historically very significant for thousands of years. It is at the junction of the Morova and the Danube. I think it was the edge of Roman Germania. In the communist era the road to it was restricted, as it was along the Austrian border (there were still lookout towers etc - I think still there). Access was permitted one day a year on (I think) Slovak national day. I was once taken on a visit by an old university professor I worked with in Bratislava. As we stood there he said "For more than 40 years I used to come here once a year, look across the Danube and think 'This is the edge of my universe'."