He studied medicine in Havana and Barcelona, earning his medical licence in 1877.
In Paris, he interned under several renowned physicians, including urologist Jean Casimir Félix Guyon (1831–1920), who served as an important influence to Albarrán's career.
In 1906 he succeeded Guyon as director of the clinic of urology at Hôpital Necker.
Albarrán's early career was largely spent in the fields of microbiology and histopathology, later switching to urology, a discipline in which he made several important contributions.
He performed the first perineal prostatectomy in France, and is credited with introducing the so-called "Albarrán lever", a device used for adjusting the movements of a cystoscope during the catheterization of the ureter.