There's only one public transportation option (that I know of) that will get you close to Lyons. I think it's this one:
http://www.rtd-denver.com/skyRide_SubHome.shtmlThe RTD DIA-to-Boulder shuttle runs from DIA to pretty much all the Boulder hotels. It used to be $12 per person one way. Book a seat in advance, arrange to be dropped off at whatever Boulder hotel is their northernmost destination (whatever their last stop is) and at that point you'll be about 15 miles from Lyons - and you'll be depending on the kindness of strangers. If you're really intrepid you could probably hitch a ride to Lyons in less than 5 minutes of standing on the side of the road. If you know anyone who's already there, they can come get you with a 30-minute round trip. A taxi from Boulder to Lyons would probably cost a small fortune.
The most difficult part of the process is obviously the ride to Lyons, and how complex that is depends on what day you arrive and whether your friends with cars in Lyons are already parked/camped and can't move their vehicle.
Book that RTD shuttle and then post a note on the ride share board on this Forum for rides from Boulder to Lyons on your arrival day and I'll bet you not only get a ride but you will make friends in the process!
Then it's up to you to figure out how to get back to the airport on Monday, although that should be easy to sort out on the ride share board.