I hope that the term "solder gun" is used as a generic name and you really don't plan to use a solder gun on your tach. Solder guns are far, far too hot for a printed circuit board and you will quickly delaminate the copper circuit pads and traces destroying the board in the process. A solder gun is a 150 watt device and is OK for BIG stuff.
Ideally, a 22.5 watt soldering iron should be used for printed circuit work but you can get away with as much as a 47 watt iron if you are careful. You also need some 24 ga rosin cored 60/40 solder (good 22 ga solder is OK, but you have to be careful and not get too much solder on the joint or you run the risk of "bridging" to adjacent traces or pads). Radio Shack is a marginal supplier for this kind of stuff, you would be better off going to a good electronics supply store (sorry about that, Radio Shack - I do use them for some things).
Resoldering all the joints on your Tach board should not be too difficult a job, just be sure to reheat the joints until the solder reflows and just touch the solder to the reheated joint. Remember that the key to a good solder joint is first of all, a good mechanical connection, and the solder is applied to the joint and NOT the soldering iron (except to "tin" it). Keep the soldering iron clean, clean it between resoldering joints, the tip should be "bright silver" in color with no rosin flux residue. A wetted paper towel is good for this, wet one and fold it into a small rectangle. Squeeze most of the water out of the towel and wipe the iron on it between resoldering joints. If you happen to wear Levi's (which I usually do) you can also wipe the iron tip on your pants leg, that also works well.
Don't miss resoldering any joints, start "left to right, top to bottom" or some such system. The Merkur (both XR4Ti and Scorpio) is full of bad solder joints, the manufacturing plant for the electronics should have had everyone take and pass a military specification soldering class before being hired for assembly work!! Good luck!! Be careful, the circuit boards are sensitive but this process very often gets good results. I have resoldered the fuel gauge, temp gauge, tach and speedometer boards in my Scorpio and this has helped the functionality of these instruments.