I have a budget resonator guitar. Is it still worth repairing?

At first, you must know the difference between repair, setup, and the upgrade.

Repair is about correcting things that were done wrong in the factory, or are damaged because of heavy/improper use or storage (ex. wrong neck angle, uneven soundwell, wrapped neck, bad scale setting, cracks etc.) In resonator guitars, repairs usually (but obviously not always) apply to the body and neck, not to the cone system which is serviced during "setup" procedures. Properly done repairs may be expensive or even inviable, considering cost of another, new or used, but undamaged body.

Setup is about putting together a roadworthy, undamaged body witch it's matching hardware (cone, bridge, nut, tuners etc). So setup is not about repairing things. You still can have a perfect body with perfect hardware sounding bad, if the setup is wrong. And - sometimes you can give a bad quality or worn off body and hardware a perfect, optimal setup to make is sound surprisingly good.

Upgrade is about replacing your stock hardware (cone, bridge, nut, tuners) with components of higher quality. It also may come with some modifications to the body itself, like soundwell routing, wedges and soundposts adjustments, etc. The good thing about upgrade is that you always can dissassemble your allready upgraded guitar, retrieve the parts and install it in another, better instrument.

What is better: a prewar National guitar or a modern replica?

This one is hard to answer. Usually (I'm not saying "always") the body and neck of prewar National (properly rebuilt and set-up after 80 years of use), will sound similar to modern, properly cared national body and neck of the same materials and costruction.
The CONE will make more difference int the sound than age of the body and neck, and what we call "vintage National sound" comes from the cone. Obviously, everything matters: body material, construction and thickness, wood type and quality of the neck. But age of the guitar is not this important. The resonator guitar is much more about the physics, tensions, angles and forces, than about the MOJO.
We may feel the MOJO by touching prewar Nationals and Dobros, searching for their history, playing them. But their cool sound is not because of it - it's just science and their unique construction.
So, if you're still unsure about the choice:
-If you are straight PLAYER, and definitely not a handyman nor a collector, get a modern guitar. I't maintaining is not as challenging and it deserves not as much attention, as in case of vintage instrument.
-If you are into collecting, aware of fragility and historical value of vintage instruments, knowing construction and weak points of resonator guitars, go for a prewar National. Apart from having a wonderful (if serviced porperly!) instrument, you'll owe a part of history and a brilliant money investment.