I saw where you mentioned you're riding a mountain bike. I hope you've put street tires on it. No sense making things harder than they are by running nobbies. That alone would make a huge difference.
Every organized ride I've participated in has rest stops every ten miles or so. Just stop and refill your bottles accordingly. Organized rides have a sort of inertia effect. Being around so many other riders seems to carry you further than you'd expect. If you can do 20 now, doing 34 with organized rest stops will be cake. For me, the biggest deal was riding with hundreds or even thousands of other riders and seeing plenty of them on clunky bikes or in similar bad physical shape as I am. You have peers out there to keep you motivated.
You'll have no problem.