Neutral and ground must be separate in a subpanel
Neutral and ground never touch, except at the equipotential bonding point in the main panel. That is a special feature with a special purpose.
The first thing you must ascertain is whether the neutral is bonded to the subpanel (fuse panel) chassis. This may have been done for the purpose of equipotential bonding. It may be separable, or may not. Sometimes the neutral bus is isolated and you can remove a screw or jumper. Otherwise you will need to obtain an accessory neutral bus that is isolated and move all the neutrals to it.
If you need a ground bus too, you may be able to get away with just clumping all grounds into a wirenut or daisy chain of wirenuts, at least for as long as it'll take to change out that fuse box. Don't use stab connectors, they are not allowed for grounds.
No ground is an island
See what I say in this answer about the danger of islanding grounds.
So don't attach grounds from a receptacle circuit to this subpanel and then stop. It is essential that the ground path go all the way back to the equipotential bond at the main panel.
Never, never, never bootleg ground
Tying ground to neutral is always a bad idea. Even if a bad old compromise exists in Code.
If the (over-tasked) neutral wire fails between the panel and the bootleg point, it energizes at line voltage everything that's supposed to be grounded. That is what makes NEMA 10 so dangerous, and it has killed. A wire simply failing should never create danger! Wires fail all the time, just look at all the "a whole bunch of outlets went out at once" questions we get.
And bootlegging ground breaks one of your essential protections: GFCI. If you bootleg ground past the GFCI -- and a device there suffers hot-ground leakage -- the GFCI has no idea and is unable to provide protection. Current is returning on ground (bad!) but the GFCI thinks it's returning on neutral like normal.
Your building is attached so you do not need a separate ground rod.
You can do as you propose: run just a ground wire from the main panel to the fuse box. The wire must be big enough for the circuit (same size wire is safe, but smaller wires are allowed for some sizes) and it can follow any viable route back to the panel or anywhere else that is grounded and whose ground path is big enough - including the house's grounding electrode system.
I'm not sure a retrofit ground would be legal at the time of subpanel replacement, but it's legal now since you still have the fuse box. Later when you upgrade to a subpanel, it should be grandfathered.
Once you establish solid ground in the fuse box, you may then retrofit ground to any circuit fed off that fuse box.