I am in the process of rebuilding a 1989 Centurion for a friend who lives near the coast. As I was stripping the bike, I noticed an inordinate amount of grit everywhere on the bike. I assume this is a combination of salt and sand.
I got to thinking that this has probably found its way into the bottom bracket that I just rebuilt in June, so I probably should repack those bearings. However, as I thought more, I began to think that converting to a sealed cartridge might be a better idea. I would like to hear your thoughts on whether replacing the old cup and cone bottom bracket with a sealed cartridge would provide substantially more protection from the elements. How hard is this conversion? I have never done this, but I am under the impression it just involves removing the spindle, bearings and sleeve and replacing them with a cartridge. Do I need to replace the cups?

The conversion is straight forward and very easy. Measure your current shell width and spindle width. All you need is a cartridge BB tool. The short answer is yes, it will eliminate crud in the bearings.

A Shimano 5 series or higher should do the trick.

Keep in mind, Shimano's sealed BBs are symmetric. Some of the older cranks back then used asymmetric spindles. Confirm this before purchasing a sealed cartridge BB. Maintain the drive side spindle spacing.

The answer to your question is yes - a cartridge unit will take the place of everything that's there now; cups, bearings, sleeve and spindle.

+1 A lot of the old ones were not symmetrical. Look closely before you buy. Keep all of the old parts as you might need them later (or be able to trade them with someone on the list for something else you need).