If you want to check for continuity, you’ll need a multimeter, preferably with a beeping continuity tester function (It beeps whenever the two points that you place the probes at have little or no resistance.). If it doesn’t have this function (as it is pretty much standard on digital multimeters, but they tend to be a bit more expensive), you can set it to any resistance test, and the needle will tip to the maximum resistance. Then, touch the probes together until the needle dips to the lowest point. Whenever it dips to the lowest point in the circuit, it means that there is little to no resistance between two points. The advantage of the beeping continuity meter is that you don’t have to look at it, it will beep when there’s continuity, but these multimeters can run a bit more expensive. Worth it if you plan on using it a lot, but for one project, you can buy whatever works.
Basically, now that you know how to check continuity, you’ll need to find the microswitch soldered to the left input (it will be the switch that is on the right side, because when the top of the joystick tilts left, the base of it shifts right, if this doesn’t make sense, tilt it to the left, hold it at left, and then flip the case over to see which microswitch is being pushed down). When it’s active, there should be little or no resistance, meaning there should be continuity between the two metal points that come from the switch. Also, when probing for resistance or continuity, the red (+V/? max) or the black (COM) probes don’t matter. While it’s active, if there’s no continuity the microswitch is broken. If not, then check for continuity between the wire that connects the microswitch to the PCB at both points to see what the problem is.
EDIT: When you say the diagonals don’t work, does that mean Right-up and Right-down don’t work either? Is right working a-okay?