I definitely always get this in the early season. I did my 5th day riding yesterday, and I've gained an hour of good leg time since my first day. It's a little worse this year because the new Mystery chair allows me to get about twice as many runs in in any given time frame. (Awwwwww, poor donutz
What killed me last year was my first experience with a big mountain (Whistler). Runs that take 20 minutes if you're really hoofing it are gonna hurt.
I try to break up my day by working on different things, but once the legs start to give out, there's not a lot of options.
As far as improving your stamina -- I don't have bad knees so I can't give any advice to compensate for that. But ideally holding a high horse stance (feet slightly wider than shoulder width, knees bent and pushed as far apart as you can manage, thighs at about 45 degrees, torso as vertical as possible) for as long as you can is surprisingly good for the thighs. To put that in perspective, to get your first belt (at least at the schools I've been to) you have to hold the horse for 2 minutes. Doesn't sound like much, until you try it. Extra points for getting your thighs parallel to the ground but I imagine that'd be really hard on the knees.