woolite in the washing machine. cold water on the gentlest cycle. To dry, carefully lay your wet shirt onto a towel. Roll the towel up with the shirt. Wring the hell out of it. The water will transfer to the towel, and rolling it in the towel also prevents it from stretching. Now hang and air dry overnight. The next day, your shirt is clean.
woolite in the sink by hand. I've washed many, many wool clothing items in sinks around the world while backpacking. Fill sink with cold water (if it's cold outside, coldish/lukewarmish water is okay, but not hot). Gently use your hands to get the dirt off. the water will turn dark. Drain sink and repeat. Now fill sink with water again and put the woolite in (doesn't take much. for a typical bathroom sink, about a tablespoon or two). Wash thoroughly with your hands, grab parts of the shirt and rub together all over. The water will turn dark again. Drain sink. Fill again, wash again without woolite. The water will turn dark again. Repeat until the water runs clear. Your shirt is now clean. To dry, see above.
The good thing about wool is that it resists bad odors and bacteria naturally. So, you can actually keep wearing a good wool shit without cleaning it often. Even while backpacking in the desert and mountains, I only cleaned my wool shirts ever 4 days or so.