If you've already built everything with detail brushes, then you have a different problem entirely. Vis should, then, run very fast. Near instantly, even. Just how big IS this skybox? (Also, you should do skies in JA using skyportals; not a big box around the level.) The BSP stage splits the world on all world-brush planes, hint brush planes, and also at regular chunk sizes. So your big giant box is cut into lots of really tiny chunks. There's an option in GTK radiant to render chunk boundaries in the 2d views. Generally, you should try to align macro-geometry to chunk boundaries. Increasing the chunk size for outdoor levels will definitely help. You should try that first. IIRC, the command-line option is -chunksize, and the default size is 1024. You can measure it in gtkradiant if show chunk boundaries is turned on. For big under-ground vis blocking brushes, they should just be caulked world brushes.
About areaportals; they only work if they entirely separate two areas of the world. There can be no connection between two areas that doesn't go through an areaportal. Also, if you built the building using detail brushes, but then plug the door with an areaportal, the areaportal isn't going to do anything - the detail brushes don't block vis.
About hint brushes; in sof2map and q3map, they create a high-priority BSP split plane down the largest axis of the brush (They should be axis-aligned). In q3map2 (Which I hope you are using) they create a high-priority BSP split plane along every surface of the brush. You use them to control the BSP process, so you can create more effecient BSP trees.
There's a plugin that comes with gtkradiant for viewing BSP portals. Load it up and play around with hint brushes.