May 1st - Labour Day in Paris

Transport systems will be running a slightly lower frequency than weekdays, but still working. Most shops will be shut. Museums and attractions will be open but may be busier than usual. Restaurants may or may not be open, if there's a particular one you really want to eat at I would contact them directly to ask.

On May 1 the museums are closed. This is a good day for a walking tour, and walking about to take in the festivities of the day wherever you are. Check Paris Time Out weekly publication, comes out on Wednesdays, for latest updates on events, etc.

There are three days a year when most major tourist sites are closed: Dec 25, Jan 1 and May 1. The Eiffel Tower is operating on May 1 and I believe the Musee Jacquemart Andre. The parks will be open and there will be plenty of places to get coffee or meals, but much will be closed. Paris unlike London never closes down its public transport for holidays so buses and metros will be running on their holiday schedule and transport will not be a problem. A good day for neighborhood walks or park visits. I am sure the Seine cruises will also be operating.

We did a Paris Walks tour--the Pere La Chaise cemetery tour. Otherwise just walked through different neighborhoods, the Jewish quarter in the Marais, Pont Neuf, the point of Ile de la Cite, etc. There's plenty to see even if you don't go into a museum. Plenty of restaurants open too.

Arrival day: Take a walking tour of the neighborhood your lodging is in.
If you're up to it join a Paris Walks or Discover Walks tour in the afternoon.

Consider a 2 day Museum pass for Tuesday & Wednesday and pre plan how you best want to use it. Thursday: Look at a bike tour or a themed walking tour. Maybe have a picnic in a garden or get some goodies for an evening picnic at Champ du Mars Parc.
The boat companies are running on Thursday, too.

I forgot to mention "Context Paris Tours".

It appears that Luxembourg Gardens is open that day, so in the worst case you find a market, grab a picnic, and relax.

I think Monet's Garden in Gervney will be open. It's about an hour away.

Enjoy a "photo safari" and any special events you may run into.