Is this something you will do a few times here and there, or is it something that will become part of a standard workflow and done repeatedly? If the former, this seems like a fairly straightforward 2-step process: Select By Attribute followed by Select By Location. Use Select By Attribute to select the polygon(s) in question, and then Select By Location to find the points the selected polygon intersects. If you want to create new data sets or update data, i.e., not just select data, Dan's Spatial Join suggestion might work better.