Don't know anything about the winery.
I'd plan for two eventualities:
Good weather: See at least one of the small towns (your choice: coastal or hill) accessible via public transportation from Nice. Coastal transport is much easier, but I think you can get to the hilly places like Eze and St. Paul-de-Vence by bus even in April. No place was dead when I was in the area May 1 - 15 last year, so I think April will be fine. You'll need to pick up a guidebook (doesn't have to be the current edition) to read about the many, many attractive options. In particular, if you like art, it seems that practically every town has at least one worthwhile art museum (check closing days, though weekends are usually fine). If you want to maximize the territory covered, you can consider a one-day bus tour, but I'd want one that focuses on the hill villages since the coast spots can probably be reached faster by train.
I have a special fondness for Menton for its old-town architecture, liveliness (read "lots of tourists") and two really fine gardens. However, those gardens are both on hillsides, and the footing could be pretty treacherous if it has rained recently. But really, there are so many lovely places; get a guidebook and see what grabs you.
Bad weather: Stay in Nice and concentrate on the museums (Matisse, Chagall, Asian Art, Naïve Art, plus many more). Verify hours of operation; some may be closed on Sunday. When the weather improves, hit a market.
No matter the weather, you'll definitely want some time for wandering through Nice's Old Town. If the weather's nice, a walk along the Promenade des Anglais is fun.