Before you can get Blair to partner up with you, you will need to obtain three items: A barrette, a bandanna and a hat. To get each of these, you must have at least four hearts filled for the characters Alexis, Syrus and Zane. Once you get their hearts up, each character will give you whatever item they have.
Once you have all three items, you have to find Blair. She appears in the cafeteria between 7 and 8 p.m. Given that you only have 1 hour of game time to find her, proper time management is a must. Once you do track her down, give her the three items, and she will disappear.
Once Blair disappears she will no longer be found in the cafeteria. To find her again, head to the Slifer red dorm. She'll only be there during school hours, from 8 a.m. to 4 p.m. She'll be standing in the upper left corner of the dorm. Talk to her, and if you can manage to defeat her in a duel, she'll become one of your available partners for tag matches.