Born in Rome, Pulcini began karting professionally in 2012, and came 7th in the KF3 category in the Italian ACI Karting Championship, won the ROK Cup International Final in the Junior ROK class. He rounded off the season participating in the 23° Trofeo Andrea Margutti - KF3 category, coming in 22nd.
In 2013, he came runner-up to the Italian Championship in the KF3 category, and made his WSK debut, competing in the WSK Super Master Series and the WSK Euro Series in the KF Junior, coming in 9th and 6th.
In 2015, Pulcini switched to Euroformula Open full-time with DAV Racing, scoring his maiden victory at the first race at the Red Bull Ring and finished ninth in the standings. He also partook in the first round of the 2015 Auto GP Series.
The following year he switched to Campos Racing.[2] With them, he took seven victories, three pole positions and eight fastest laps to secure the Euroformula Open title.
GP3 Series
In November 2016, it was announced that Pulcini would reunite with Campos for GP3 Series post-season testing at Yas Marina.[3] In January 2017, Pulcini signed to race with Arden International.[4] A second place in the first round at Catalunya would end up being his only points-scoring finish and the Italian ended up 14th in the championship.
For 2018 Pulcini moved to Campos Racing in what would be the final season of the GP3 Series. His results were much better, and after two race wins and three further podiums Pulcini finished fourth in the drivers' standings.