Can white rum be used in this recipe? If so, would the measurements be the same?

I would add that not all white rums are created equal. Those in the Puerto Rican style (including those from the USVI) are designed to be nearly flavorless. Others, specifically pot-stilled versions, are quite the opposite. Examples of the latter: Jamaican Myers's Platinum, Hawaiian Sammy's Beach Bar.

I think this is the recipe you're referring to:
http://food52.com/recipes/2332-rum-raisin-rice-pudding

Yes, you can use light (white) rum - use the same amount. The dish will have a slightly lighter color and taste but it will be fine. Bon appetit.