My question about this Akathist is this:
Why does it personify, and even say the cross is worthy of Worship?
For why the word "worship" is used, I would point you to Shanghaiski's post. Whether the Akathist, in its modern English form, should continue using the word worship is another matter, but certainly the reason it is used is because of the more archaic meaning of worship = giving due worth to anything, divine or not.
As for personifying the Cross, and addressing prayers directly to it, that's a good question. I've thought about it before, and subsequently replied to someone else about the subject on another forum. This was part of my post:
The concept of personifying non-sensate things is not unknown in the Church, even the early Church, and of the Old Testament righteous before them. St Paul, repeated by St John Chrysostom, addresses death: "Where, o death, is your victory? Where, o death, is your sting?"
. "Death", or "Hades", are often depicted on Icons of the Resurrection in a human form, bound by the angels; the seas and the River Jordan are often shown in Theophany icons in human form:
... these depictions are based upon the Psalm verse: “the sea saw and fled, the Jordan turned back”
Now, both of these examples can be thought of as "literary", because Paul is not praying to "death", even though he is addressing it; conversely, David is praying (it is a Psalm), but only references "the sea" and "the Jordan", and doesn't directly address them. However, David does
address non-sentient things in other parts of the Psalms, that is, in prayer:
1 I was glad when they said unto me, Let us go into the house of the LORD.
2 Our feet shall stand within thy gates, O Jerusalem.
3 Jerusalem is builded as a city that is compact together:
4 Whither the tribes go up, the tribes of the LORD, unto the testimony of Israel, to give thanks unto the name of the LORD.
5 For there are set thrones of judgment, the thrones of the house of David.
6 Pray for the peace of Jerusalem: they shall prosper that love thee.
7 Peace be within thy walls, and prosperity within thy palaces.
8 For my brethren and companions' sakes, I will now say, Peace be within thee.
9 Because of the house of the LORD our God I will seek thy good.
This is clearly a Psalm - a prayer - addressing the walled-city of Jerusalem: not the people of Jerusalem, but the city itself. There are other Psalms which directly address inanimate objects (the Lauds - Psalm 148-150 - being another example), but I think Psalm 122 is the best "prototype" of a Prayer/Hymn to a non-sensate thing. If David can sing a Psalm to the buildings, palaces and gates of a city, how much more can we sing praises to the wood through which Christ has saved us!
Finally, we have to think of the context. Just as Psalm 122 appears among a whole section of Psalmody which was always intended to be sung as one (the "Psalms of Ascent" - Psalms 120-134), so to do hymns to the Cross appear as only a part of our daily office. We don't address the Cross apart from Our Lord Who suffered upon it, and turned such a cruel instrument of torture into a sign of victory. Even an Akathist hymn is always "inserted" into another service like Vespers or Compline, which contain prayers to Christ and the Holy Trinity. Below is an example from the vespers service before the feast of the Life-Giving Cross:http://www.anastasis.org.uk/cross_vespers.htm
As you can see, there are a number of hymns to the Cross, but also to Christ our God too.
As I said, most of the above I wrote some time ago, but it comes from my own wondering about why the Cross is addressed as a person in prayer (and the life-giving tomb too, in other Akathists). This is what put my mind at rest regarding the subject; I hope it helps you too.
ps: for reference, I have not used LXX numbering of Psalms