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I have been informed by Joe Louis Arena representative Rebbecca, that there are no shuttles from Flint into the Arena during redwings games which are provided by the Arena.
International Taxi & Shuttle Company will take anywhere from 1 person to 6 people for a flat rate of $150 each way.(NNN) NNN-NNNN1942 S Dort Hwy, Flint, MI
I don't know if that fits within your budget however, it is the only commercially available option I was able to locate.
Just a shot in the dark... If you perhaps might have friends in Flint with a young adult son or daughter, perhaps you might be able to ask them if they might like to earn some cash for the day. I am sure it would be much less than $300!