If buying an adapter to play Famicom carts on your NES is not an option, you read about the mapper and make a decision based on the information you can find about it. If the mapper is similar enough to one that's easy to find in the US, you can modify the hardware and recreate the board the game needs. The other is t mapper-hack the game to a more common mapper that has the same features of the original mapper.
If you look at the info about
The Goonies, you'll see that it has 32KB of PRG-ROM (i.e. no bankswitching) and 16KB of CHR-ROM (2 8KB pages). If you read about mapper 87 (I used
Disch's docs), you'll see that it only really does one thing: it selects a CHR-ROM page when its only register is written to. This is very similar to a very common mapper,
CNROM. All you have to do is play the game in FCEUX, open the debugger and set a breakpoint for writes to $6000-$7FFF (this is the area where mapper 87 expects register writes). This way you can find the part(s) of the program that use the mapper. You have to hack these parts to use CNROM instead of the original mapper.
Note that CNROM has bus conflicts, which means that in order to bankswitch successfully, mapper writes must be made to ROM locations that contain the same value as the one being written, and this takes a bit more code. This means that everywhere you see an "STA $6000" instruction (or similar, like "STX $7000", or any store in the $6000-$7FFF range) it has to be replaced by "JSR $XXXX". $XXXX is an address where there are free byte of ROM you can use. In that location you will have to put a small subroutine that bankswitches CHR-ROM the way CNROM does it.