I am nearly 38, so not exactly old, but I felt old before starting cycling 2 years ago. Wasn't particularly overweight, but half lost about a stone and a half. The issue was fitness and lethargy.
Now I feel so much fitter and can't wait to get out there and ride.
The point of my rambling is, not whether I can manage to cycle as I get older, but can I manage not to. Age doesn't matter, as you can be so much fitter in your 80's if you get exercise, than in your 30's if you don't do anything at all!
I plan to keep doing this until I drop, no question about it!