I just re-read the original post and saw that you have a whole year before the tour, which is much better than I originally thought.
I expect it can be done, but you'll have a lot to learn and a lot of miles to put in between now and then. I know you want the bike to be cheap, but you'll just end up discouraged if it's too cheap because you didn't take time to learn the importance of various aspects. Whatever you have now, go ahead and start riding lots, and take time to educate yourself about what bike you'll want for the actual tour, instead of rushing to buy and then being sorry as you find out later that there were important considerations you missed in your hurry. You may find that the new bike is still not a very big part of the overall cost when you count the other equipment, the camping, food, etc..
Even if you can't ride through the winter, riding a lot now will get you a head start for next spring and summer as you learn to become more efficient and handle the bike better. That experience stays with you even if you get out of shape in the winter.
Hang around the forums and ask questions and learn. If you get books like David suggested, be aware that bikes and equipment have changed substantially in the last decade or two, and more has been learned about fitness/nutrition/health on the bike. IOW, if you get a book from 1980, don't exclude other materials just because you haven't finished digesting it yet.