The pattern “the first person (to do something)” (or same with any other noun in place of “person”) normally takes to + infinitive, not the -ing form. I think this is because the first time something is done is viewed as one discrete action, not a continuous action going on over time.
Both those sentences are grammatical, but the combinations do not seem tremendously likely to me (albeit not impossible).
“He was the person to …” seems to need something a bit more dramatic and immediate, and/or possibly with more emphasis on a succession of events, as in “When the call came, he was the person to step forward and volunteer.”
“He was the person to …” is possibly more often used with “he / the person” as the object of the verb rather than the subject. For example, “He was the person to ask” would often mean that one should have asked him.
“He was the person verb-ing …” needs a context in which the continuous tense is more appropriate. For example, “He was the person shouting” or “He was the person walking in the middle of the road.”