How do Unitarians understand 1 Corinthians 1:2?

It is very simple: Jesus came in the Father's name. It is a shared name.

I am come in my Father's name, and ye receive me not: if another shall come in his own name, him ye will receive. (John 5:43, KJV)

Jesus' name is the only name whereby we must be saved.

¹⁰Be it known unto you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom ye crucified, whom God raised from the dead, even by him doth this man stand here before you whole. ¹¹This is the stone which was set at nought of you builders, which is become the head of the corner. ¹²Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved. (Acts 4:10-12, KJV)

This name was already told us in prophecy before Jesus was born in Bethlehem.

For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counsellor, The mighty God, The everlasting Father, The Prince of Peace. (Isaiah 9:6, KJV)

The name given to Jesus was assigned to him before his birth.

And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins. (Matthew 1:21, KJV)

And, behold, thou shalt conceive in thy womb, and bring forth a son, and shalt call his name JESUS. (Luke 1:31, KJV)

This "name" is the name of both Jesus (the Son) and the Father. This is the "name" into which we are to be baptized (see Matthew 28:19) and by which we are to be saved (see Acts 4:10-12).

Disclaimer: I am a non-trinitarian, but not a member of the Universalist or Unitarian church.

This has nothing to do with a unitarian, binitarian, or trinitarian perspective.

All Christians believe that it is Jesus's sacrifice that pays the penalty for their sins. This is true whether they believe that Jesus was created by the Father (unitarian), is an eternally existing being co-equal with the Father (binitarian), or is one of three persons that comprise God (trinitarian).

Consider:

Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.
— Acts 4:12

This says that to request salvation, one must explicitly ask that Jesus's sacrifice be used to pay the outstanding penalty for one's sins.

So when 1 Corinthians 1:2 says:

… all that in every place call upon the name of Jesus Christ our Lord …

it need not mean anything more than an invocation of his name with respect to his sacrifice. There is no assumption implied concerning his origin, divinity, or relationship to the Father.

The answer depends on the type of Unitarian one means. Christian Unitarians often believe that salvation involves accepting Jesus and "calling on his name," while rejecting the idea of the Trinity. However, in 1961 the American Unitarian Association merged with the Universalist Church of America to form the Unitarian Universalist Association. Often referred to simply as Unitarians, members of the UUA usually do not insist that one must accept Jesus as one's personal savior. The denomination affirms that:

We have no shared creed. Our shared covenant (our seven Principles) supports “the free and responsible search for truth and meaning.” Though Unitarianism and Universalism were both liberal Christian traditions, this responsible search has led us to embrace diverse teachings from Eastern and Western religions and philosophies.

Christian Unitarians would have no problem accepting what Paul says in 1 Corinthians 1:2. But Unitarian Universalists would tend to disagree, insisting that there are many paths to the truth.