Totally doable, and a good idea.
We know that "Obsolete computers running a space ship" works - because it is a summary of NASAs space shuttle program.
NASA needs parts no one makes anymore.
So to keep the shuttles flying, the space agency has begun trolling the Internet -- including Yahoo and eBay -- to find replacement parts for electronic gear that would strike a home computer user as primitive.
Source: For Parts, NASA Boldly Goes . . . on eBay - New York Times
So yes you can fly into space with decades old computer tech - because we did it. Vietnam-war-era tech flew until 2011.
Parts are obviously going to be hard - but if you have a source of parts (like old tech to salvage for parts), you'll be able to do it.
These parts are also easier to fabricate than modern parts - setting up a semiconductor facility is hard work, but making 1970's parts in an improvised way is a lot easier to do than making 2040 tech. We can 3d print a transistor using 2020 tech, so a 1970's-equiverlant IC could feasibly be 3d printed using a 2040-era hobby 3d printer.
"Obsolete computers running a space ship to avoid an AI taking it over" is not just a good idea, there's precedent in sci-fi.
I would suggest watching the 2003 remake of Battlestar Galactica, with a particular focus on their strategic use of 50+ year old tech for all critical systems as a way of preventing infiltration by AI.
A laptop computer used by crew on BattleStar Galactica:
A tablet computer used by crew on a similar ship in the same fleet just a few months prior to AI's attack:
Key to the strategic technical regression was the lack of any computer networking; for example FTL jumps were plotted on one computer, and coordinates entered into the drives control computer by an operator with a keyboard reading off a screen. By having an operator sit between the two computers selectively transferring data known to be valid the AI can't abuse that wire link to spread itself.
You may not need to go all the way to magnetic tape cassettes - a slightly more modern computer with no networking may / should / (up to you) be immune to AI infection, but if the best parts you can make are tape cassette tech, that's what you'd fly with, and thus you're setting is quite feasible.