Shield arduino for electrical interference
You can do several things to reduce the effects of interference on the pin2 input. There is no one answer of how to prevent EMI causing problems, each system is different and each system needs a different solution. But here's some general tips for you to try:


Reduce interference at the source. This is the best, but it's not always possible.  You could check that the switch on the solder station isn't too worn, but you're probably not going to gain anything here.  I've written it anyway, because it's a good general rule.
Reduce R1. Smaller pull-down resistors mean more current available and less susceptibility to EMI. If you were using 100k, I would definitely suggest a smaller value, but you might still see some gain from going below 10K.
Filter/debounce. The usual approach to debouncing switches, with a capacitor, can help reject brief transient EMI. It's also generally a good idea.
Sheilding.  Put everything inside a metal enclosure, and connect the enclosure to ground.
Wiring. Avoid long lengths of wire, and especially avoid long lengths of wire outside the enclosure. Where long wires are unavoidable, group them together to reduce loop area. If S1 needs to be outside the box, connect it using twisted pair (or even better, shielded twisted pair). Connect the shield to the enclosure.