Alnæs studied music first in Oslo with Iver Holter, then in Leipzig with Carl Reinecke and, after the première of his first symphony in 1896, in Berlin with Julius Ruthardt.
From 1895 to 1907 Alnæs served as an organist in Drammen, and then for many years played the organ in several churches and conducted choirs in Oslo. He helped found the Norwegian Society of Composers. Alnæs wrote music in a late Romantic style.
His output included two symphonies, one set of symphonic variations, a piano concerto, pieces for piano, chorale preludes for organ, choral works, and art songs (in Norwegian, romanser).