One site suggests that a refrigerator has 5900 megajoules of energy embodied in it. That's accurate enough for napkin math, so we'll run with it. 1 kWh = 3.6 megajoules, so 1638 kWh per refrigerator.
https://energyeducation.ca/encyclopedia/Appliance#Embodied_e...
Two hops upthread, the commenter says "280kWh/year" for their 1938 fridge. No clue what kind of power they use, but we can calculate the recoup cost of a replacement without knowing it.
It looks like modern refrigerators are considered about 3x as efficient as pre-70s refrigerators, so let's say that a straight swap of a size-equivalent refrigerator of average efficiency will cost 100kWh/year.
https://www.energystar.gov/index.cfm?fuseaction=refrig.calcu...
Spending 1638 kWh on a new refrigerator will save 160kWh/year, so the break even point in carbon terms occurs after 10 years have passed.
This is all very bad napkin math done in crayon. If you want better figures, you'll need to know your own refrigerator runtime figures, which means measuring your fridge's power usage for a standard week of use (if you maintain a very stable indoor temperature through heating/cooling) or for a year (if you do not), and you'll need to know the energy saved per year of your current fridge, and etc.
ps. Apparently they have a fridge from 90 years ago that's still running. If you replace one modern fridge with another and only save 20 kWh/year, it will take you 90 years to recoup that cost. The 10 year figure is only possible because their 1938 fridge is so inefficient.
pps. Owner of said 1938 fridge, I intend no disrespect. This DOE paper talks in detail about the changes that improved efficiency – inventing high-efficiency compressors, changing the shape of the drain pipe (1% efficiency gain!), and replacing the refrigerant and insulation materials with more efficient ones. https://www.nrel.gov/docs/legosti/old/7281.pdf