That is correct because there's no such thing or place as purgatory. Did his execution for theft expiate the temporal punishments due all his OTHER sins or did his simple faith in Jesus Christ as Savior satisfy the sin debt God demands?
If the day of Pentecost and the scene in the House of Cornelius are any indication, believing has immediate SALVATION/TRANSFORMATION results, not entry onto a yo-yo trail of sacraments.
False dichotomy. In his case, the purification occurred when he believed in Jesus, expressed remorse for his sins, and did the good work of defending the innocent and rebuking the guilty. He died a good Catholic death.