How can I properly replace the hose on this older Schwinn pump?

A search online turned up nothing for a pump that old. If you have a model number or something else that identifies the pump, that may help get better answers. There are plenty of replacement hoses out there, but the photos don't really show a closeup of the threaded connection to the pump base. The only thing I can think of, would be to take the pump and the hose connection piece to an automotive shop where they can make their own high pressure hose fittings for vehicle applications like air conditioning type hoses. You would obviously need a smaller diameter hose, but it's possible they have smaller hoses for other high pressure applications. They may already have a threaded fitting that will work, or know where to get one. If they have or can get the fitting, then the correct diameter and length of hose should be easy. This would likely cost twice as much or more than you paid for the pump, but if it's a favorite, it may be worth it to you.

I think that the size of that thread is called, "1/16 pipe thread" Find local places that make custom hydraulic hoses and verify the thread. Then ask whether they have a fitting that is 1/16 male thread by hose barb. Then you could eliminate all of the 30-year-old hose.

Another option would be to very carefully cut off the piece of metal that crimps the existing hose onto the threaded part, and then reuse the threaded part by crimping the new hose onto it. The downside is that if it doesn't work, you can no longer even go on as you are, because you will no longer have any of the old hose left at all to join to. If you can find another hose that fits first, like maybe a one from a similar age frame pump (which would be too short to just use as is), that would be safer.

Yet another option (which could also be used if you try the option above and it fails) - pick one of the adapters that came with the new hose, and make the pump fit it, either by tapping it out to a larger thread that fits a larger adapter thread, or tapping it out to a larger thread and fitting a helicoil. Downside - if the new thread is too much larger, the base of the pump will break. (And if it's not larger enough, it won't hold the new adapter securely. You could cross thread it and then use epoxy, but that's getting pretty desperate.)

This looks similar. You could whack off the head and hopefully the hose inlet to the pump is the right size.
http://www.sears.com/custom-building-products-replacement-pump-hose-20inch/p-SPM6180332701?prdNo=10&blockNo=10&blockType=G10

It is listed as 3/16" hose connector.

It's rather simple, just buy another hose in the length you need. There's common lengths of you can get one custom made at a hydraulic hose shop possibly.

A Google search of "Bicycle pump replacement hose" will pull up dozens and dozens of hoses in a variety of lengths, end fittings and colors.

I just ordered a Silca replacement hose and Schrader end for the same pump. Silca also makes a presta valve adapter. Check out the Silca website or Modern Bike.