The flight is by KLM from Singapore to Nice (via Amsterdam), depart at 01.15 from Singapore, arrive at Amsterdam at 07.50
If that's what it says on your itinerary, I suppose that is the correct arrival time. I got my information from the website of Schiphol Airport and I'm not quite sure how often the long-term timetables are updated there.
no bus from plane to terminal
Larger aircraft such as those operating on the SIN-AMS route are, very exceptional cases aside, always parked on one of the gates with passenger boarding bridges (jet bridge/jetway/passenger walkway, whatever you call them), so you don't need to get onto a bus to reach the terminal building.
As for the passport control, note that I'm using Schengen Visa to enter Netherlands and France. Correct me if I'm wrong here, but I think I still need to pass through the passport control in that case?
You will go through passport control upon arrival at Amsterdam, which is your entry point to the Schengen Area. There are no (regular) passport checks as such for intra-Schengen flights, regardless of your visa status. Security staff and (at the departure gate) KLM staff will want to see your passport too, but that's just to make sure the name on your boarding pass matches that in your passport.
When you get back to the airport, you can head straight to the entrance to the Schengen part of the airport (Departures 1), get in line for the security check (queues can be long there, but you should be able to get through in 15-20 minutes even when it's crowded), show your boarding pass to security staff, get through security and walk to the gate. More likely than not, it will take you around 10 minutes to get to the departure gate from the security checkpoint.
I read that there is a museum inside the airport
There's a museum between the E and F piers, but don't be too enthusiastic: you can have a good look at the exhibition in 15-20 minutes, if you take your time.
will it be heavy raining, as it is probably early spring, thus I won't be able to walk around the city?
As Blagger pointed out, that's impossible to say. Technically it's still winter, I remember a quite horrible day at Schiphol Airport in early March 2005 when heavy snow caused lots of cancellations and massive delays, but you could just as well arrive to find sunny weather and a pleasant 10°C (not quite Singapore, I know). Or it rains or drizzles and everything looks grey and depressing. No-one can predict that (but no matter how the weather will be, locals will be walking around the city).
