PCI-E's link negotiation sorts out throttling the x16 card to a x1 link assuming you can connect it physically. Adapters/converters make this easy and are pretty simple to find online. You can also mod the slot or the card if you're feeling brave (not a great idea). Similar adapters to run x16 cards on a x1 link through ExpressCard can also be found (laptop with 5850--yes please?). Powering the card could be a bit hit or miss and there is a bit of a performance hit to think about.
The converters aren't very large, but most cards are probably too big for the case. As I said, this isn't a great upgrade path to take for rather obvious reasons, but if you were going to do it you could find a card with a shorter, non-standard PCB (Gigabyte makes tons of them) that might fit with the right aftermarket cooler.
Still, I doubt OP's system has a PSU that would safely power anything worth buying. Your recommendation to save for a new system is sound. I only posted so that OP's aware of the (rather esoteric) option.