His family were Latter-day Saints and named his after the Book of Mormon prophet Alma.
In 1866 as a result of Utah's Black Hawk War the leaders of the Church of Jesus Christ of Latter-day Saints (LDS Church) instructed the settlers along the upper Weber River to move together in a central location.
At various times Eldredge worked in railroad building, brick making and as a store owner.
In 1878 he was elected Summit County prosecutor and a member of the Utah Territorial Board of Trade.
From 1877 to 1901 Eldredge served as a counselor to William W. Cluff in the presidency of the Summit Stake of the LDS Church.