After spending 5 seasons in the Yankees minor league system the Mets drafted right-handed pitcher Mike Draper in the Rule 5 draft in 1993. He made his Major League debut on April 10, 1993 by pitching two innings of relief in the Mets 6-3 loss to the Houston Astros; he allowed 1 run and 4 hits. Draper would go on to pitch in 29 games for the Mets, all but one out of the bullpen, and finished the season with a 1-1 record and 4.25 ERA. His one start occurred in his last appearance in a Met uniform on August 7, he lasted just 3 innings and allowed 3 runs and 5 hits but the Mets still managed to win the game 10-8 over the Pittsburgh Pirates.
Following the season, the Mets released Draper and he was picked up by the San Diego Padres but would not appear in another Major League game.
While with the Mets Draper wore number 47.