Afraid your idea of a minimal cost big-bore .416 or .458 Winchester Magnum rifle is sorta confusing.
Unless you can do most of the metal and wood work yourself, here are the challenges:
1. Both are hot, "high intensity cartridges, and cheap actions are NOT the way to go with either caliber.
2. The most readily available "cheap action" will be a surplus M1917 Enfield or an 1898 Mauser action. Both require lots of machine work to feed and fire the magnums. a good action will probably run you $150 minimum.
3. A reputable gunsmith will have the action magnafluxed or x-rayed for integrity (no cracks, fissures, etc.) and hardness tested. $40-$60 lab fee.
4. A barrel blank in either caliber will set you back a minimum of $150. No cheapskating here, again, because they are hot calibers.
5. Machine work for fiting barrel to action, alteration of boltface, magazine lips, extractor, bending bolt, drill & tap for scope mount, scope mount, scope, etc. polish & blue. Minimum $30/hour labour fee.
6. Custom stock inlet, fit, checker, & finish.
7. Few alternatives available: both bores are too large to re-bore/re-chamber from an existing common rifle caliber. Few sporting rifle barrels (except heavy varmit barrels) think enough to handle a re-bore.
When done, you are looking at more than the cost of a factory rifle in either caliber. I'm guessing, but I'd say it's the rare gunsmith that can pull this project off for less than $700 or $800, minimum.
Still want that "economy rifle"?
Curious John