MitchMan, sorry I didn't get back more quickly.
First, I'm not a professional repair guy - just setting expectations - so anything I'm saying here is a swag, but like most of the readers here I've had to struggle with my own 'old boat' challenges.
So, yes, I think we both agree that the 'white stuff' was added by the PO and covers some more serious problem.
To repair you'd need to do the following
1. Remove shaft. This implies removing the engine coupling (can be difficult!) and probably removing the prop. If you take off the prop you'll likely be able to swing it to the side enough to not have to drop the rudder, but I don't know Catalinas well enough to say that for sure.
2. Chip out the 'white stuff'. The issue here is going to be access. In my old Gulfstar 37 I could only get to this area when I was curled up in a cockpit locker. Hopefully your access is better, but think about what tools you'll need and how you'll get two hands in place to use them.
3. Remove and replace the stern tube - this is likely damaged already (what the PO was trying to repair) but chipping out the white stuff will almost certainly damage it fatally. Removing that brass tube (hopefully what it is) will not be a trivial task. You should be able to look at the exterior end of the tube to see what it is - I'm worried that's it galvanized iron pipe and it's rusted in the hull, causing the leak. Other Catalina owner can certainly comment on this before you pull the boat.
So now, how long? First, my experience with my own estimates are that I'm regularly at least 1/2 too low, so I'd add something onto what I say below. Given that it's a boat (and an older one) you'll want to adjust for what you already may know that I don't.
1. Remove shaft - if everything pops apart easily then this is a few hours work. If the prop or coupling is frozen onto the shaft then it'll be alot more. Again, on my 37 I never got the prop off - too many years in salt water without proper maintenance. If it doesn't come easily I'd consider paying a boatyard for this since they have tools and experience you don't have and if they mess something up (bend the shaft or the prop) then they have some responsibility to make it right. So I'd say plan two days of part-time effort, maybe one day if you have the right tools and parts, etc.
2. Remove old white stuff - this depends completely on what it is. If it's fiberglass and resin/epoxy then you almost have to cut/grind it out - it won't chip away. If it's hardened putty or caulk you may be able to chip or shave it away with an air or electric chisel. You won't be able to get it all until the old stern tube is removed, but you'll be cleaning up the area to prep for the new repair. Access to the area also colors this estimate, but I'd guess you could easily spend a couple of days on this as well, more if it's difficult to access.
3. Replace stern tube - once you have the area clean and prepped you're down to the easy part. I'd get the right brass (?) tube and make sure it fits into the hole in the hull with a small bit of play. Flare the hole on both ends quite a bit (maybe more than you think) to allow the new fiberglass/epoxy some good surface to bond to. The hull should be quite thick here (several inches?) so don't worry too much about structural integrity unless there's existing cracks (which would be a whole other problem). Apply caulk to the stern tube and the inside of the hole and slide the tube in so that the center part of the tube is bedded in caulk. I'd do this from the inside so the other end of the tube can be cleaned - easier to clean it up outside. Slide the shaft back in place securely (can be done before sliding the new tube in) and use wedges to center the tube in the hole, then glass it in. I'd use a small amount of epoxy and shredded fiberglass to fill the very bottom of the flare, then build up layers of cloth and epoxy until it's fair with the hull. This will be very hard inside - you'll be stuffing it in with your fingers and/or a stick under the tube. But as said above, this won't be structurally important so you should be OK. Once it's all hardened up you just (JUST! LOL) need to reassemble. Because the 'glassing has to happen in one shot, I'd plan two days for this - one day to line everything up and get ready, maybe place the tube and shaft, then one day for the glass work. Probably a third day for reassembly.
While you're doing this you should also replace the cutlass bearing in the strut - it will almost certainly need it, and that can be a day or two itself, depending on how difficult the old one is to get out.
So in total, with my estimate, you're looking at a week of full days, if you're doing it on weekends/evening only it'll be longer. A yard may be able to do it more quickly, but you'll need to trust/review their work.
I'm sure others in the forum can add/correct alot of this. Also make sure the shaft is REALLY checked for runout/burning/groove while it's out - be a shame to do all this and still have the stuffing box leak.
Hope this helps...
Marty