Put raisins, poppy seeds, sugar, and rolls into a large bowl and stir to combine; transfer to a 10" × 13" casserole dish and spread out evenly; set aside.
2. Put milk into a medium pot, bring just to a boil over medium-high heat, then pour over bread mixture. Toss gently to mix without breaking up the rolls. Let cool slightly, then cover with plastic wrap, and refrigerate overnight.
3. When ready to serve, remove dish from refrigerator and scoop pudding into small bowls. Drizzle with rum and serve immediately.