How to fix water leak coming from pressure gauge/valve on water tank?
The hole is to provide atmospheric pressure inside as a reference for the gauge.  That is, the pressure indicated is relative to ambient pressure.  Or, in other words, the gauge indicates how much more pressure there is inside the pipe than outside the gauge.

If the gauge is leaking through the air vent, it is time to replace the gauge.  The internal mechanism has failed, probably by rusting through or otherwise breached the pressure-sensing diaphragm.

Freezing conditions could well stress the gauge possibly due to a little ice build up.

Yes, this is something most anyone can do.  Turn off the pump and relieve the pressure in the line (open a faucet).  Using a wrench, vicegrip, or similar tool, unscrew the gauge assembly (gauge plus its short input pipe).  Using the tool, grasp the gauge's pipe which comes out below it.  There should be a four- or six-sided grip (turn it counter-clockwise as seen from above.

If it has been installed for a long time (years), it will be reluctant to turn at first.  A longer handled tool will provide more torque, but take care to provide counter-torque to the pipe it is screwed into to prevent damaging other parts of the plumbing.  (Provide counter-torque with another wrench or tool attached to the plumbing near where the gauge is screwed into.  If the wrenches are about 90&deg; relative to each other, it is convenient and efficient to twist them opposite to each other so that the water pipe experiences greatly lessened torque.)

Once you have the old gauge assembly out, take it to the parts store for a suitable replacement.  Consider using a fine wire brush to clean the seat (the threaded hole the gauge is screwed into); Teflon tape makes it easier to insert the new gauge.

I wouldn't wait long; do it in the next few days.  It will leak more and more the longer you wait.