Stamina takes time and work. Going on trail rides and letting him trot up hill will definitely build some muscles. In the arena do a good workout, probably stop once you got him swettin pretty good or when he gets tired. My DAILY work out for my horse is:
Walk around arena once
Work on bending for at least 3 laps.
Start slow trotting(hes a reining horse so he has the very little trot) and do that for 2 laps going each way
Then do the extended trot each way 3 to 4 times(and bending during this helps with the neck)
Then lope around the arena 2 times each way then start doin figure 8's with flying lead changes Or ill split the arena and do smaller circles for awhile.
Then I walk him out until he is cooled off.
And I have to admit my horse is pretty buff :]
Also I wouldnt do all that off the bat work him up to it, iv been doin this with my horse for a long time he has the stamina for it he is barely swetting when im done with all that.