Unfortunately, Genesis 2:23 is not translatable in a single verse. It is so compact and laden with irony, subtle humor, and multiple meanings.
The choice that you make for translating הפעם depends on which nuance of the verse you think is most important.
If you read with the MT's readers marks (taamim), as does the targum attributed to Onkelos, and Rashi, then the reading is something like
And the person [adam] said, "This time, a bone from my bones and flesh from my flesh..."
similar to the use of הפעם in Judges 15:3 and Genesis 29:35.
If you read against the MT's readers marks, as does the targum attributed to Jonathan ben Uziel and the Sforno and some modern translations like the Cambridge New English Bible, then the reading is something like:
And the person [adam] said "This [the woman], at last, is a bone of my bones, and flesh of my flesh..."
similar to the use of הפעם in Genesis 46:30 and Judges 16:18.
The MT reading makes it sound like Adam is saying "This time God got it right...", which makes Adam sound impudent and is ironic in juxtaposition with Adam blaming God in Genesis 3:12, "The woman that you gave me for a companion...". This appears to be a caustic comment about the way men commonly view women; bone of my bone, flesh of my flesh, until something goes wrong, and then the woman and God are to blame.
Another problem with the MT reading is why Adam says "this time" at all. To what previous times is Adam referring? Rashi, who reads with the MT, answers this by referring to verse 20, and explaining (based on Midrashic sources,1) that Adam had tried to mate, unsuccessfully, with all of the animals prior God bringing Eve to him. This fanciful midrash is supported by the language of "bringing to" in verse 22, which is suggestive of the Hebrew euphemism for a meeting with a mating partner.
Other commentators following the targum of Jonathan ben Uziel suggest that the "this time" refers to the means of creation of Eve, that this time she was created from a person's (Adam's) rib, but next time an thereafter women would be created by normal procreation.
The reading against the MT reader's marks, in which זאת (this) refers to Eve, rather than "this time", gives הפעם the meaning of "at last". This reading makes Adam's apparent joy with his companion humorous or silly, in view of the events that are about to happen and Adam's sudden denial of responsibility.
Either way that you read the verse, הפעם is key to understanding the irony of the message.
1. Recorded in Babylonian Talmud tractate Yevamoth page 63 side a.