Nappi was born in Rome. After signing with Urbe Tevere in 1982, he went on to play for several other Italian clubs, starting his professional footballing career with A.C. Cesena in 1983, remaining with the club until 1986, also spending two seasons on loan with Ravenna (1984–85, and 1985–86). He moved to Arezzo for a season in 1987, before moving to Genoa in 1988. During his time with Genoa (1988–99), he notably won the 1988–89 Serie B title, helping the club to obtain Serie A promotion, and also winning the 1995–96 Anglo-Italian Cup; he was also sent on loan to various clubs during his time with Genoa: Brescia (1989 and 1994–95), Fiorentina (1989–91), Udinese (1991–92), and SPAL (1992–93). He ended his career in 2006, after a season with U.S.D. Sestri Levante 1919. During his time with Fiorentina, he played alongside Roberto Baggio and also Stefano Borgonovo, helping the club to reach the 1990 UEFA Cup Final, where they were defeated by rivals Juventus. Following his retirement, he worked as a coach.