No other games will add to that circuit board itself (at least no outside some incredibly esoteric wizardry that might possibly be done to change what game that circuit board was running).
The Neo Geo will have a motherboard that looks roughly similar to your Tekken Tag one, but has a cartridge slot for Neo Geo cartridges.
An adapter to specifically adapt Neo Geo to your Tekken Tag machine would be way more complicated then just making the couple of wiring changes needed.
Both circuit boards use a 56 pin main connector (called a JAMMA connector).
Your Tekken Tag has a second connector that wires up buttons 4 and 5 (and the second speaker, although this is often not used.
While a Neo Geo circuit board (of the one slot variety), has button 4 wired up to a normally unused wire on the jamma connector. The multislot variety also wires the sound differently.
If you want it easy with no wiring changes then you are going to want to just buy jamma boards (game circuit boards) for games that will plug into your wiring with no changes.
Check games on
http://www.arcade-museum.com to see if they match up to your game cabinet. Here are the details you need to match up to make sure it will work in your cabinet with your wiring.
Orientation: Horizontal
Type: Raster: Standard Resolution
Joystick: 8-way (or 2-way)
Buttons: 3 (or less)
Conversion Class: JAMMA
arcade-museum.com can be wrong about details from time to time, so it wouldn't hurt to ask here about specific boards you might want before you buy them (that site actually lists Tekken Tag as being medium resolution but I know for a fact that it isn't, as Tekken Tag Tournament is the circuit board I run on my monitor test bench.