You've jumped past the key first step of the analysis: how does the new house affect your FIRE plans? Even without upgrades, a house is going to be more expensive than an apartment, and an old one even moreso. Would you be buying a house if you weren't going to get this one? If you planned to buy a house, would you buy this one? Getting a "free" property is fantastic, but you shouldn't let it change your long-term plans, and it doesn't convert an unaffordable lifestyle into an affordable one.
Your baseline option is to sell the house and then use the proceeds toward your life goals. Maybe that's hitting FIRE as quickly as possible; in that case, the money goes into investments. OTOH, if your life goals have always included owning a home, and this one suits you, and you can afford the ongoing costs, then congrats! Keep the house, check that goal off your list.
Fundamentally, I think of owning a home as a consumption decision. Sure, there is long-term value there that many consumer goods don't have -- but in order to realize that value, you actually have to sell the home, which most of us don't want to do. Most of us buy homes because we want nice places to live. And when you're already living somewhere reasonable, any money you spend above that figure is luxury spending.* So evaluate the new home from that perspective: how much more will it cost than you are currently spending, how much will it improve your life, and is the increase in luxury worth that increase in cost?
The first thing you need to do is figure out what your ongoing homeownership costs will be -- not the remodel, but taxes, maintenance, electricity/heat, periodic roof replacements, changes in commute costs, etc. Make sure that fits within your budget. If you're lucky and are paying a lot in rent, it could even save you money, which would be awesome -- but don't just assume that. Do the research, do the math.
Assuming it fits within your budget, the next step is to take a second look at your EF and decide whether you're still comfortable with that current level. For example, if you decided that you want your EF to cover 3 months of expenses, and those expenses go up, then the EF needs to go up as well. That money needs to be put away first, preferably before you're in the house.
Once you have the EF squared away, you need to go back to your larger financial plan. How many years do you have until you want to FIRE? How much $$$ will you need to put away each month to stay on track? Ideally, that amount will go into your long-term investments, and you can use any extra to save up for home repairs. Of course, for most of us, we don't make enough money to both stay on the original FIRE plan and pay for major home renovations. So if you're in that situation, look at how a variety of scenarios will affect your FIRE date -- then decide what balance is right for you. Maybe you can do the renovations in full if you're willing to push back FIRE 10 years. If that's too long, then maybe you can space out the renos, or DIY more, or cut back the scope of the plans, and extend your FIRE date only 7 years, or 5, or 2. The point is to do the analysis yourself and figure out what compromise feels right for you.
Old houses: yeah, they can be expensive. If I were you, I'd get a full home inspection, as if you were buying the property. The last thing you want to do is move into this great gift that needs $100K in foundation repairs and a new roof. And if someone is living there, you should be able to get information about what their utility bills are. You can also research options your local utility may have to improve energy efficiency, cut the costs on installing solar, or whatever else you can find.
From my experience, the issue is less with insulation than it is with sealing the sills and joists and such. We have an old house, and we injected insulation into the walls, but the floors on both the first and second floors were always cold as ice in winter, because that kind of blown-in insulation can't fill the gaps and cracks you get in the sills (same thing around windows, btw). We got "lucky" (and I put that in quotes because "lucky" = house fire that made us take the house down to the studs) and had to do re-siding and new drywall, and so we added foam insulation and house wrap and made a point of sealing every gap we could find. But that's not something that's easy to do with a fully-functioning house. What you can do is get an IR camera (I think there's even a phone app) and go around looking for cold spots in winter, then tackle those one by one to see if you can seal them up somehow, either from the inside or outside.
*To be clear: there's not a damn thing wrong with luxury spending. All of us do it in one way or another. The point is to recognize it for what it is, vs. trying to rationalize it as a "need," or as a "great investment opportunity," or some other story we use to justify doing what we wanted to do anyway.
