The last time I went to Limon (the only time I went there) from San Jose was in 2005. I was studying abroad and we took a bus, but we went out at night in Limon and it was perfectly safe. I don't know much about the actual drive from SJ to Limon, but I assume it's safe.
I actually think Costa Rica in general is really safe. Just like anywhere else in the world including the U.S., you have to be smart about things, know if there is a "bad/rough" part of town just like all bigger cities have, etc. I've never felt unsafe in Costa Rica, and I studied abroad there for 5 weeks in 2005, then was in the Peace Corps there for 7 months in 2007/2008. I only took public transport during those 8-9 months I spent there over those couple years. After 2008 I have gone back four times to visit the families and have rented a car all four times.
+1 on the public transport/bus option. If you want to rent, I have found a local rental car company that has good prices and great customer service, with no "gotchas" or secret fees. They are called "Vamos-Rent-A-Car." They have an office in San Jose right near the airport. My wife and I just were there in November. For 4-5 days I think we paid around $120 for a small SUV. Gas is pretty expensive though, if you covert from liters to gallons and colones to USD it was a little over $4/gallon I think.
Regarding cell phone coverage, we brought our i-phone and downloaded the free app "waze" which is basically just google maps but the locals say it works better than google maps. It was great for us, and I didn't notice any gaps in cell phone coverage. We drove about 2-3 hours/day all over the country over those four days and the app worked great for navigation. If you don't want to bring your own phone, Vamos Rent a Car will let you "rent" a local phone for $5/day, although I don't know if it's a smart phone with the waze app or just a flip phone for calls. (I rented it once but it was 2008 or 2009 so it was just a little flip phone for local calls/texts.)