If I'm understanding correctly, there are 2 heating systems or boilers. One in your half that's on it's last leg and failing fast and one in his half that's presumably working fine. Rather than replace the broiler system in your half he wants to extend the pipes from his side to yours so both halves will share a system. He'll do all the plumbing work, but he wants you to pay $1000 for your portion of the fuel.
You don't feel you will use $1000 worth of fuel and regardless, you don't want to or can't pay upfront.
What he's proposing sounds no different to me than rental properties where utilities are included. Except in those situations rather than a lump sum amount, the rent is higher to compensate for the expected utility use. Stupid move for a landlord, at least IME. Too risky, especially with today's price swings.
You could propose a lesser amount based off your more usual fuel consumption. Offer as payments or a rental increase. Since you're already planning your move, which could be part of why he doesn't want to install a whole new unit in your half, you could say to heck with it and go with electric heaters or a window heating unit. He may also decide to heck with it and provide you with an electric unit.
Something to keep in mind, were it to go to court, I would expect you would both end up spending more than the $1000 and you would still be out more $ either for heating fuel or electricity.
*************
Just re-read and saw that you had offered weekly payments. I do think that's fair and would be a good route if you both can agree on the amount. The utility company could help you with avg usage amounts. Did he say why he didn't want to except payments? Is there some compelling reason that you two could compromise on? How much are you paying now for fuel?
Honestly, I would probably go the electric route (I'm assuming here you are responsible for your electric usage) since you like it on the cooler side and go with a portable system. There are package window AC/Heat units. I know a lot of people that heat entirely with one or two electric fireplaces. Get something you can take them with you when you move and either use in your new place or sell.