Alabi grew up in Zaria, Kaduna state of Nigeria before moving to Clermont, Florida at the age of 17. He attended high school at Montverde Academy in Montverde, Florida where he was coached by Kevin Sutton. Alabi was an all-state performer and helped lead Montverde Academy to a perfect 30–0 record during his senior season. He attended the same high school as Minnesota's Luc Mbah a Moute. Alabi started for the Nigerian Junior National Team at the 2007 Nike All-American camp. He also helped Nigeria qualify for the 2007 FIBA Under-19 World Championship. He was selected for the 2007 Nike Hoop Summit, representing the World Select Team, where he led all players in blocked shots.
After breaking his leg 10 games into his freshman year, Alabi had to take a medical redshirt. As a redshirt freshman, he earned All-ACC Freshman Team honors by averaging 8.4 points per game and leading the ACC in blocks with 2.1 per game. For his great shot-blocking ability, he was named to the ACC All-Defensive team in 2008–2009.
As a redshirt sophomore, Alabi ranked 26th in the nation in blocked shots with an average of 2.39 while increasing his scoring average to 11.7 points per game. On April 23, 2010, he declared himself eligible for the 2010 NBA draft.
On June 24, 2010, Alabi was drafted by the Dallas Mavericks with the 50th pick, and traded to the Toronto Raptors for cash considerations.
On July 8, 2010, he signed a contract with the Toronto Raptors. The Raptors assigned him to the Erie BayHawks on November 15, 2010. He was recalled on December 9, 2010, sent back to Erie on January 6, 2011, and recalled once again by the Raptors on January 14, 2011. The Raptors assigned him to the BayHawks for a third time on March 9, 2011. Afterwards, Alabi was recalled back up to Toronto for a third time on April 5, 2011.
On January 4, 2012, Alabi was assigned to the Bakersfield Jam of the D-League. He was recalled on January 22, 2012. On April 26, 2012, against the New Jersey Nets, Alabi recorded career highs of 11 points, 19 rebounds and 3 blocks in 40 minutes in the final game of the regular season.
On October 1, 2012, Solomon signed with the New Orleans Hornets. However, he was released on October 27.
On December 28, 2012, Alabi joined the Idaho Stampede. He was released on March 1, 2013.
On March 21, 2013, he signed with the Greek first division basketball club Ikaros Kallitheas B.C.
On September 28, 2013, Alabi signed with the Philadelphia 76ers. However, he was waived on October 5. He later signed with the Yulon Dinos of Taiwan for the 2013–14 season.
In January 2015, Alabi signed with the Barako Bull Energy for the 2015 PBA Commissioner's Cup.
On June 27, 2010, it was revealed that Alabi had Hepatitis B, which was the reason for him sliding so far in the NBA Draft.