How much easier is cycling compared to walking or running? I'm thinking of taking part in a cycling event, I'd like to do something that's about as challenging as running a half marathon (13 miles / 21 km). Approximately what distance should I be looking at?

I think in principal this could be worked out with a heart-rate monitor (if I had one): 


Walk or run for (say) 15 minutes, note the heart rate and the
distance traveled 
Cycle for the same length of time with the same
heart rate and note the distance traveled
I know calorie counts vary quite a bit, but a rough ballpark for cycling at a 20 mph pace is 450 calories per 15 minutes according to this calorie calculator.

Running a 6 minute mile is expected to burn a similar 450 calorie count on the same calculator with the same settings.

So, if you do the math, and the calculator is assumed to be reasonably accurate, then 5 miles on a bike at 20mph, is roughly equivalent to 1 mile run at a very fast pace.

Running a half marathon, which is 13 miles, and ignoring any difference due to endurance of effort, 13 miles at full pace running could be assumed to be a similar effort to 65 miles on a bike, in a very general sense. 

This is dependent on fitness, body weight, and many other factors.

The best method to work it out would be to measure power expended, but I don't know how to do that for running, and to be accurate, you'd need to do it yourself, and for the full distance of the half marathon.