Leaving the airport, look for a sign that says "26 San Juan" and take that exit. Follow signs to San Juan and the "parrot route". After about 4 miles, take the right exit to "22 Bayamon and Caguas". It goes through "Tunel Minillas" and leaves you on the expressway 22. Stick to expressway 22 / parrot. There are two major divides that can be hard in traffic: A couple miles after the tunnel, 22 is the right three lanes; then a few miles after that, Route 5 divides off to the left. Stick to 22 (there be dragons). Once you get to the first toll booth, you are fairly safe. Drive about an hour, and get off in Arecibo at Route 129 (it's the third Arecibo exit) towards Arecibo. Turn right at the light, and then right again at the next light, leaving you on 651. Stay on that road until you see signs for the observatory (in about 20 minutes). There's one intersection where the road number changes from 651 to 635. Go straight through the intersection.
Once you see the signs, it's pretty clear. The first sign is at a junction with 626: You want to turn right, and stay on 635. Then go about 1/2 mile and turn left at the cemetery onto 625. Follow 625 to the end (2.5 km), bear left at "El Observatorio" restaurant, and arrive at the gate. There's a guard at the gate who can help you 24/7.
It's very hard to get a decent Puerto Rico map anywhere but on the island. Most gas stations here have fine ones. If you get lost, ask anybody, or call the observatory: (787) 878-2612. The telescope operator (extension 211) always speaks English.