If you're running on a modern computer (and as long as you don't have any glaring inefficiencies), I don't see why you wouldn't finish under a few seconds seconds.
Why use the 0'th element for a boolean was visited? Technically it should work, but that's taking away from the OO nature of Java. Create a node class which contains a boolean isVisited and a list of nodes that node is connected to. Personally, I wouldn't recommend using an array because it will make building your graph a big pain, try using a built-in list type such as the ArrayList or LinkedList, these will make building your graph much easier (actually, technically you could use any collection which allows you to iterate through all the elements).
In Djikstra's algorithm, you actually need the min distance from the start node to every other node. I kind of forgot about that earlier, so that would just be another field inside your node class. When you've visited every node, you can just pick a target node and print out the min distance field of that node.
edit: oh, one last thing (hopefully) is that you need to initialize the initial min distance from the start node to be infinity (or something really large), otherwise there's a chance you'll get a bogus result.