It will all depend on what kind of bike you have and how long you want to stay. If your bike has over 350cc in engine capacity, you could face a lot of problems to get it into the country. You will need a special clearance from the Defence Ministry. Usually, as a tourist, and for temporary import, it could be done, but the red tape could be lengthy as well. As you would only temporarily import the bike, and re-export again, no import duties are to be paid.
If you want to stay just a short while, your biggest disadvantage would be transport cost. Sri Lanka is (still) an island, so you'll have to ship it in and ship it out again. So it might be wiser to send to bike to wherever you would start your 'continental' part of the Asian trip, and rent a bike while you're in Sri Lanka.
The other guys gave you good advice: head straight to Negombo if you want to rent a bike. It's close to the airport and has much more rental facilities (as a tourist spot) than Colombo. About 25 km around Colombo, traffic is hell also. I know, I'm travelling from Negombo to Colombo every day!
If you need more advice, or any help, just ask, and I'll see what I can do.