You probably could if you really wanted to.

The place where they rent the bicycles is a little way up Elizabeth Street, you'd cycle down Elizabeth Street, through the pedestrian mall (might have to walk it there, if there were a lot of people about), and straight down to the water, or just about to the water. That's maybe 800 metres. Then you'd turn left and follow the bicycle paths to the official start of the Cycleway. About 1 km or so. Then you'd cycle the cycleway out to the MONA turnoff. Approx. 11.5 km MONA is about 800 metres off the Cycleway. All up, round trip, it would be about 26 km, and relatively flat because the Cycleway is relatively flat.

There is quite a bit of bicycle parking at MONA.

We cycled there once (with our own bicycles) and spent an afternoon wandering around. It's ... different.