No bikes at the airport, but you can get a free 90 minutes ticket good for use on all public transportation (bus, train, tram) at the airport; the ticket machines are next to the green exits. This will get you to one of the free bike rental stations. Take any train to the Geneva main station (first stop, ~6 times an hour), where the main bike rental place is.

As the English link appears not to work (is French spoken in southern Ontario? ), here is the deal:
- free rentals only from 4/25 to 10/25
- four hours max, can be returned at any of 5 locations
- operating hours 9am-7pm, 8am-9pm for main site (Montbrillant, behind main train station)
- regular rentals also available for longer periods

The area around Geneva is actually quite nice to bike through, once you get into the countryside. It is easy to explore since you are hemmed in by the lake and French border, with the Saleve and Jura mountains as further visual roadblocks, so it is nearly impossible to get lost. Biking season here is roughly April to October, May to September for short sleeves.
As for Geneva proper, it is small and the 4 hour limitation will not be a problem. Nothing really spectacular to see though (my jaded native's point of view...) A visit to the northern shore of lake Geneva could be nice (by bike if so inclined, one can always load the bike on a local train if tired), with some of the more scenic routes/stops being around Nyon, Rolle, and the stretch past Lausanne to around Montreux.

What is your budget for accomodations? I doubt anyone would visit GVA for a mattress run, and a savvy mileage runner probably would want reasonably priced lodgings. There are some inexpensive-for-Switzerland options if you are willing to go slightly off the beaten path.