I see it as similar to what would happen if a viking longboat suddenly appeared in modern times with a crew that decided they'd like to become part of their country's navy. Assume for sake of argument that this viking longboat is endowed with a magical stave that allows it to instantaneously transport itself to whatever point on the globe the captain chooses. This magical stave is imprinted on the hull of the boat itself and cannot be removed or transferred.

We would re-enforce the wooden hull of the boat with steel. No doubt a nuclear reactor would be far too massive and complex, so we would likely install gas turbine engines, or perhaps maybe even an electrical propulsion system, space permitting. Radar systems would be installed, but we couldn't use a phased-array radar system; it's dependent on the geometry of the ship's superstructure. There wouldn't be room for a vertically-launched missile system, but an older deck-mounted missile launch system would certainly be possible. A 20mm deck gun would work just fine, but sea-based artillery is out of the question. A 5-inch gun requires far too much supporting equipment below decks. Communications would be no problem; install a mast on the newly steel-reinforced deck and place a small-ish transceiver somewhere, and the boat now has the same high-bandwidth comms afforded to modern ships.

Computing power would be somewhat of a problem. Modern ships rely on large spaces full specialized military-grade purpose-built computing systems. We could likely fit a server rack with a few high-powered Xeon servers somewhere on the ship, but it wouldn't have the same total-ship integration that something like an Aegis cruiser or destroyer has. Processing power might be the same (if not better), but the overall topology of the system would be vastly different, and likely more limited in what sort of inputs the ship's "main computer" is able to receive and process.

I would suspect that a 23rd century starship in the 32nd century would be in a very similar scenario. As long as the ship is still considered spaceworthy, many modern advancements could be installed or easily modified to fit the 900-year-old ship. Because the unique capabilities of the USS Discovery are inextricably tied to the ship itself and it's crew, however, there will always be limitations on what technology can be installed without altering the structure of the ship to the point that the spore drive no longer functions. That's part of the reason why I think the spore drive in the Discovery and a magical stave on a viking longboat are a good comparison. Both have "magical" elements and may or may not rely on the ship itself being set up in such a way to allow these magical elements to function. Change the ship too much, and it's no longer the same ship.

That does bring up an interesting argument regarding the Mk2 spore drive that was used on Booker's ship. If there is a way to create a spore drive that can be operated on any modern starship, is there really any reason for the 900-year-old Discovery to still be in active service? Seems to me that a 32nd-century starship with Stamets onboard would serve the same function, only better.