It sounds as if you will already be in France before May 1, but just in case that is the day you fly in, be aware that on May 1, 2017, there was zero public transportation running from the airport into the city, and precious few taxis waiting. I was very lucky to bump into another arrival who wasn't being picked up, and we shared a taxi into town.
The weather was gorgeous (really the nicest day of the 2 weeks I spent in the area; May weather can be a bit unsettled), but the streets were very quiet in my neighborhood near the Nice Ville RR station. Perphaps a lot of the locals were out of town. I think many of the restaurants were closed as well. That may have only been at lunchtime; I had crashed by dinnertime. There were still places to eat, of course.
Although I'm not too optimistic, you can go to the individual museums' websites and check for information on holiday hours. You may get lucky.
Trains will be running on a holiday schedule, but it won't be a problem to move east or west along the Riviera. Eze and St-Paul-de-Vence require careful checking ahead of time, because those require buses, which will likely be very infrequent or non-existent on the holiday.
What I'd do is plan to visit at least one or two of the atmospheric coastal towns where wandering around outside looking at the historic buildings is a large part of the attraction. I liked Menton, which also has a couple of very impressive gardens of the pay-to-enter variety. Both are built on the sides of hills, so should not be attempted by those with mobility issues. According to its website, Serre de la Madone will be open on the holiday. It is not within walking distance of the Menton RR station (you want the main station, the first one you come to in Menton when approaching from Nice), and I think bus service on May 1 is doubtful. Plan to take a taxi. Note the phone number so you can call for pick-up when it's time to return to the city; probably someone at the garden would call for you. There were several taxis outside the station when I visited, but I can't promise what will happen on May 1. The second large garden, Jardin Val Rahmeh, is closed on May 1.
You could also walk partway around Cap d'Antibes or Cap Ferrat. The paths are not paved all the way, and you'll need good shoes, or you could go just part of the way around. You can end up quite far from shelter, so don't attempt this if heavy rain is threatened.
Hey, I found something else that will be open: The Villa Ephrussi de Rothschild on Cap Ferrat. The villa is interesting and the gardens are very, very nice. I have my doubts as to whether the local buses will be running, so you might need to walk from the Beaulieu-sur-Mer train station (or there might be taxis hanging around). I walked but don't remember how far it was beyond "not very close". The nearby Villa Kerylos is not open.