I climb (mountaineer) as part of my job. For a very modest price (<$100) you can build a very capable and safe climbing rig that will get you to the top of your mast and keep you there in relative comfort for long enough to complete repairs.

The technique is called jumaring (or jugging), and is accomplished with a harness (though I have used a "riggers belt" in the past) 2 ascenders, a few carabiners, and a few 2m lengths of 1" tubular webbing. I will not go into details as to the technique, as I do not want to encourage anyone to attempt it without proper instruction, but if you were to go to your local climbing center and explain what you were trying to do you could be quite proficient in a few hours.

Properly performed the technique is not strenous, and is very safe, utilizing redundant ascenders. It will not damage your halyard (assuming that you aren't climbing the mast several times a day), and is easily accomplished alone in all conditions (though in heavy seas you may need an additional piece of webbing to hold you to the mast to prevent being hurled about).

PS - In order to get back down you will need a descender of some sort. There are too many types to list...