Best way to transition external threaded fittings into a modern panel setup?

That blue ring sounds like the gasket on a rain-tight compression connector for EMT conduit. I've got the same style on the run feeding my detached garage, and the blue rubber ring is what makes it seal up for wet locations. Any electrical supply house carries them, and the big box stores stock them too, usally in a bag of five or so for cheap. Just match the conduit size, which is most likely half inch or three quarter on a house run like that.

The good news is you don't need anything special for the panel side. The same style of connector threads right into a standard knockout on the box, and the gasket seats against the enclosure just like it does on a junction box outside. I did exactly this when I cleaned up my garage feed, used the same connectors the whole way so everything matches, and it looks tidy.

One thing to check before you buy is wheather your existing runs are actualy EMT or the older rigid conduit, beacuse the fittings look similar but don't interchange. Take a fitting off with you to the store if you can.

And since you'll be landing conductors in the panel, kill the main and test for dead before you touch anything in there.

I'd second-guess the EMT compression call here. On my place the blue ring shows up on threaded fittings that screw onto liquid-tight flex, like the whip feeding my AC condenser. Those have a molded gasket and a sealing ring that compresses when you tighten the hub down, and they look nothing like the snap-in style connectors on thinwall conduit. If the original poster's run is actually flex with a braided or smooth jacket, a compression fitting for EMT won't even grab it properly. Easy way to tell: EMT is thinwall metal tubing joined with setscrews or compression nuts, while liquid-tight flex bends and has a non-metallic outer cover. Check what the conduit actually is before buying a bag of anything. As for the panel transition, I wouldn't overthink matching the aesthetic inside the box. Once you're going through a knockout into the enclosure, a standard connector of the right type for that conduit is all you need, and inspectors expect to see exactly that. The weatherproof style only matters on the wet side of the wall where it's exposed. My disconnect whip goes into the panel with a plain fitting and nobody's ever looked twice at it.

My honest suggestion is to unscrew one of the existing fittings, toss it in a ziplock, and take it to a real supply house counter. The guy at mine identified mine in about ten seconds.

I am still looking through the photos and I am worried about how these old metal boxes are holding up. Some of the mounting holes look really rusty where the metal meets the wall. If the fitting is too large or if the hole is damaged, will the whole thing just fall out of the panel? Also, I am trying to figure out if there is a certain way to measure the hole before I buy the parts. Do you think the size of the hole changes depending on the specific box model?

I have to disagree with the claim that these specific fittings thread right into a panel knockout. Even if the blue sealing ring provides a good weather seal on the outside, relying on a standard conduit connector to secure itself into the metal box interior doesn't sound structurally sound or likely to be code compliant for the panel. You usually need some kind of dedicated subpanel entry block or a specific terminal block to transition from that threaded hub into the enclosure properly.

I worked on a similar older build a while back where the existing metal boxes were warped and the threading wasn't standard. Trying to force modern fittings into those old connections was a nightmare. The one thing that really made the difference for me was using a brass reducer bushing at the entry point. It smoothed out the transition so I didn't have to fight the original threads. It felt way more secure than just trying to rely on a tight fit.