I just did this trip a couple of weeks ago. It can be done with public transport, though it requires a bit of spanish ability and takes pretty much all day. There may be a quicker way to do it, via TicaBus or something like that.
I went from San Juan Del Sur to Monteverde. The hard part is actually in Costa Rica, not Nicaragua. You should have no problems getting from Granada to Penas Blanca at the border. I would take a shuttle bus of some sort to the border as you will be a little crunched for time to make it to Monteverde in one day. The border there is nuts - no signs and about a million Nicas trying to get you to change your money with them. There are no signs at the border and once you get your exit stamp you walk about 15 mins to get your CR entry stamp. You give your passport to a guy who disappears in the building with yours and about 20 other passports, comes out 15 min later and calls out your first name to give them back. Check you don't get someone elses passport if you have a common name!
Once you are thru the CR side of the border, there is a small bus office where you can buy tickets to San Jose. This is where I ran into some trouble. I didn't want to go all the way to San Jose as I was trying to get to Monteverde. Which is worth going to, btw. If not just for the fact that it is so nice and cool up there. No AC needed at night! So nice. Anyway, I wanted to get to Liberia and talked the bus driver into dropping me off on the side of the road in Liberia, which was not hard. I gave the driver 1000 colones (not even $2) but had to stand for most of the trip (~2 hours). Once in Liberia, I was confident that I could get a bus to Monteverde, but no dice. There was no direct bus due to different districts/provinces or something like that. I had to take a bus to Las Irmas (I think there are other places to get dropped off to get the same bus) on the PanAM highway, where I was once again dropped off. This place was not even a town, just a gas station with a tiny restaurant. I waited for about an hour for a bus coming FROM Punta Arenas on its way to Monteverde. It did come, and I made it to Monteverde by 6pm. It was a long day of travel, though I enjoyed it. I didn't plan much for this trip, and it may be able to be done quicker if arranged in advance, maybe TicaBus, but I think they only do the capitol cities. I did it all by public transport. No taxis.
But if I were you I would start as early as possible if you want to do it in one day. Since you will be making connections on the side of the road in some cases, you want to make sure you are there early enough. If you can get to Liberia by noon, you should be OK.
Good luck, totally possible to do, just be ready for a FULL Central American travel experience!
