Yes, there's probably a lot of snow in late March!
The first screenshot below shows the height profile of the Bergen - Voss bike ride. As you can see, the highest point is 464 m.a.s.l at Kvamskogen. The second screenshot shows the weather and snow depth in 2015 at Kvamskogen. In late March 2015 there was approximately 150 cm of snow.
There's a fair chance the road is free from snow, but the normal temperature is 0 degrees, and it rains/snows a lot, so it's 50/50.
The bike ride is 170 km and hilly, and the weather can be quite harsh. I strongly recommend you to not take this trip in late March, unless you're very well trained and prepared for many hours in cold windy and wet conditions.
It's a very nice ride in the summer time, so I highly recommend you try to do the trip another time!