There are two approaches most consider for this, and varies with what they do.
They have some kind of X in 1 game and just want one of them to be autobooted as that is what they want to play (sometimes also making a version for the other games so they can put it on their flash cart/emulator list and use that to select instead).
In most cases this is mostly just finding whatever command is issued to jump to a new location by the selection menu and putting that command so it happens earlier in the boot process (sometimes the menu does something necessary but not always).
While this does mean getting hands deep in the code you can probably learn it fairly quickly, especially for the NES with something like http://fceux.com/web/help/CodeDataLogger.html
(it logs what code is run, you would then sit at the menu for a while, move up and down, do any options... basically everything but launch the game. Then launch the game, among the pieces of new code run will be a jump to where you want, construct that jump and put it in a place earlier in the ROM).
Whether this will fix your issue with the emulator (I would generally say get a better emulator but eh) I don't know for, as the others mentioned, this is probably more of a mapper issue but more on that shortly.
Extracting games so they run on lesser flash carts and emulators or so you can use the space for your hacks is the other part of it.
The basic NES hardware is not very good, does not hold much storage and otherwise is rather limiting. To that end NES game developers made various so called mappers to expand the storage, memory, audio and more on the cartridges http://wiki.nesdev.com/w/index.php/List_of_mappers
There are lists of official ones from games sanctioned by Nintendo or otherwise used in notable games (see things like Tengen Tetris). However when homebrew makers, Hong Kong based pirates and the like appeared they often constructed their own mappers that were unknown to emulators and flash carts unless the authors of said same figured out how it worked and included it, which few did or stuck around to add all the dozens of new ones cooked up each year for random small runs of things.
As mappers often are seen to literally swap sections of memory out in favour of others then it is possible sometimes to convert things to different mappers, maybe simplify proceedings and otherwise change games for more well known ones that emulators support. This will see you having to get your hands very dirty with code, learning what mappers are, why you want to use them, the sorts of things usually done by mappers and what yours do in this case to match them to something more accepted by your emulator.
If you know the game works by itself I do have to ask why you are not using that version with your emulator? If it is because your 3 in 1 dump has some kind of ROM hack applied to it then we can possibly help figure out what part of the 3 in 1 game corresponds to what part of the basic game. If it is some kind of "it is illegal unless I dump it myself" then... yeah.
On converting a ROM hack to a basic game then the idea is simple enough -- what would normally be pages/banks on the basic ROM will also be seen in the 3 in 1 game, but maybe in a different location. Most ROM hacks will not change every single byte or add masses more and so you can start by finding out where things are now -- search the game for something in the stock game that is not changed for the ROM hack.