I don't think "the real Nicaragua" exists in San Juan del Sur. It is very heavily influenced by the ex-pat community that has turned it from a fishing village to what it is today. Given the limited time frame you have, your best chance to get a sense of southern Nicaragua is to visit Granada. You pretty much pass through Granada on your way to SJdS which is 75 minutes further south. There is a wide variety of good accommodation there and plenty to see in the surrounding area. Over a couple of days you could easily visit 4 major churches and 3 museums in this "White" colonial city, the Isletes, the Masiya Volcano (National Park where you park right next to an active crater) and Laguna Apoyo, a scenic protected fresh water lake located in a crater. There are hotels and guest houses around the rim at Luguna Apoyo as well as down by the shore (20 minutes to 1/2 and hour from Granada). You might also stop by the "old market" in Masiya and take a look at the crafts there ... many will be the same ones you see in Costa Rica at twice the price (CR vendors frequently buy their crafts from Nicas). You can probably hire a taxi driver for a day for $50 to take you to these places. A tour agency will charge you 3 or 4 times this amount for a comprehensive tour.
There are direct tourist class buses to Liberia and San Jose, Costa Rica as well as tourist transfer services that you can catch to CR. You can also take a taxi to the border, walk across (.75 km, porters available) and catch another taxi on the other side. If you have not done that sort of travelling before, I suggest you catch one of the tourist buses or vans.
Edited: 9:15 pm, March 31, 2013