Had a lean-to (attached to shed) at my old house for the tractor. Worked just fine to keep the sun and weather off the tractor. Would do that again if I needed to.
When we moved to our new house, my tractor stayed outside for about a year until a new barn was built (actually, the old tractor was outside for 6 months and the new one outside for 6 months). I didn't use a tarp because I think they can be just as bad at trapping moisture as they would be to protect from rain. The only downsides were a wet seat now and then, and dealing with pine needles/leaves/etc.
I did tarp when snowstorms were coming though, so that I could easily get out to plow without having to spend a lot of time cleaning snow off the tractor. Let me tell you though, it's a pain to tarp it right when you know it's going to be windy and a lot of snow drops. No matter what I did, heavy snow would really "shrink wrap" the tarp over the tractor.
Now with a bigger barn, it's awesome. Always have a dry seat and Iike knowing the tractor is protected from weather and out of sight.