The story in Gen. 1 gives a very different account from Gen. 2. This is actually stipulated in the text. After the Gen. 1 story concludes, the Garden of Eden story begins:
This is the account of the heavens and the earth when they were
created, when the Lord God made the earth and the heavens. (Gen. 2:4)
(Note: the above can also be read as a summary of Gen. 1, but either way, it serves to distinguish the two accounts.)
The two accounts
In Gen. one there is no "Adam," there are two unnamed human beings, male and female, created in God's image. They are created on the Sixth Day after the vegetation and animals. In Gen. 2, Adam is formed from the dust prior to the vegetation, and Eve is not yet mentioned.
Now no shrub had yet appeared on the earth and no plant had yet sprung
up, for the Lord God had not sent rain on the earth and there was no
one to work the ground, but streams came up from the earth and watered
the whole surface of the ground. Then the Lord God formed a man from
the dust of the ground and breathed into his nostrils the breath of
life, and the man became a living being.
I point this out because I think these are two different stories by two different authors and it is a misstep, hermeneutically, to combine them as if they were one. For a summary of the basis for the idea of dual authorship see Documentary Hypothesis. (Note: this does not preclude IMO that both stories were inspired by God, only that it is a mistake to treat them as if they are one account.)
Was Adam alone?
Since the OP asks, however, I will point out that Adam as BOTH male and female is a very old concept. An article in the Jewish publication Forward points out:
The rabbis understood this to mean that Adam was created as an
intersexed being, a hermaphrodite; singular in one respect, plural in
another. Exactly how Adam was constituted as an intersexed being was
debated. Rabbi Jeremiah ben Elazar held that Adam was an androgyne,
while Rabbi Samuel bar Nachman held that Adam was more like conjoined
male/female twins.
So if Rabbi Samuel is correct, Adam was not alone. However, the text itself does not indicate this. In fact it depicts God as saying "It is not good that then man should be alone."
As long was were are entertaining speculation, there is also a Jewish legend that Adam had a previous wife, before Eve, namely Lilith. She did not like playing a subordinate role, either sexually or otherwise, and left him. If we include this legend, Adam was currently "alone," but he had not always been so. However, once again, this has to be considered speculation.
To take the OP question on directly: Adam was not alone in Gen. 1, but he was alone in Gen. 2, whether Eve was part of him or not.