Born in Bologna, Apicella began competing in Formula Three in 1984 and won the first two rounds of the 1986 Italian Formula Three season for Scuderia Coloni, immediately marking himself as a front-runner. He attempted the 1985 Macau Grand Prix without finishing.
Apicella graduated to International Formula 3000 in 1987 with an uncompetitive entry, scoring a single championship point with a fifth-place finish at Spa-Francorchamps. He moved to the FIRST team for 1988, arriving as a title contender, and produced a second-place finish at Monza, though the team ultimately underperformed. In 1989, with FIRST now running Reynard machinery, he collected several podium finishes on his way to fourth overall in the drivers' championship. Momentum from that year raised expectations for 1990, but his season deteriorated, ending with a disqualification at Brands Hatch. A crash at the Pau Grand Prix denied him what could have been his first outright victory. During the off-season he received test drives from both the Modena and Minardi Formula One teams.
For 1991 Apicella joined Paul Stewart Racing alongside team founder Paul Stewart himself. He finished fifth overall at season's end, adding further podium placings but still without a race win. Across his International F3000 career he started more races than any other driver who failed to win, a distinction that reflects both his longevity in the series and the margins he so often ran on.
With no International F3000 drive available for 1992, Apicella moved to Japan to compete in the Japanese Formula 3000 series with the Dome team. He finished tenth in the standings, the best result among Dome's drivers that season, and won the fifth round of the championship. Continuing to improve in 1993, he won at Sugo and finished fourth in the championship.
Strong Japanese performances earned Apicella a single-race Formula One opportunity with Eddie Jordan's team at the 1993 Italian Grand Prix — his home race. He replaced Thierry Boutsen in the Jordan 193, tested the car at Imola beforehand, and qualified in 23rd position, half a second behind teammate Rubens Barrichello. The race lasted seconds for Apicella: at the first corner of the opening lap, a multi-car collision ended his afternoon. He was classified as having completed approximately 800 metres before retirement.
Apicella is sometimes cited as having had the shortest Formula One career ever, though that record is in fact held by Ernst Loof, whose car failed before leaving the starting grid. Emanuele Naspetti replaced Apicella for the following round in Portugal.
Returning to Japan, Apicella produced the defining season of his career in 1994, winning at Mine, Suzuka, and Fuji circuits to claim the Japanese Formula 3000 title with Dome. He remained in the series as it transitioned into Formula Nippon for 1996, but reduced his Japanese activities to also contest the 24 Hours of Le Mans. He also served as Dome's chief test driver in 1996, evaluating the Dome F105 Formula One prototype alongside Shinji Nakano and Katsumi Yamamoto until the project was abandoned after the car sustained serious accident damage. His final Formula Nippon season came in 1997 with Stellar International, where a fourth place at Mine was his best result.
Apicella returned to Italy in 1999 for the Italian Formula 3000 championship, winning twice and finishing third overall. He also attempted to qualify for the International F3000 race at Spa that year without success. Back in Japan, he competed in the All Japan GT Championship and made several starts at the 24 Hours of Le Mans with the JLOC team. A heavy accident on the Mulsanne Straight during the 2007 Le Mans practice badly damaged his Lamborghini Murciélago, but he returned to compete in 2009. That event and a concurrent Super GT campaign marked the end of his professional career.
Apicella's career illustrates the fine margins that separate those who reach Formula One regularly from those who earn a single chance. A consistent front-runner across two continents and multiple championships, he won the 1994 Japanese Formula 3000 title — a genuine achievement in a competitive series — yet is primarily remembered outside Japan for a Formula One career that lasted less than a minute of racing.