What use have I for all the gold of Troy?

[Achebe]

This is a decent poem for 'Novice'.
I couldn't get used to the ambiguity in the POV - whether the speaker is in the past or present.
You'd need to cut down on the metaphors, and make them consistent.

What use have I for all the gold of Troy? ...there is a POV problem. Does the speaker have all the gold of Troy available for his use? This would be a more appropriate sentiment for one who did, whereas the speaker seems to be 
These weary eyes see not its ancient gleam. ......both 'weary eyes' and 'ancient gleam' are tired expressions, particularly 'ancient gleam' - that's both tired and a metaphor. If the poem is set in the present, then 'ancient gleam' is ok, but if in the aftermath of the war, confusing.
Where once the verdant grass of Hope did grow ...cut out the 'did' and rephrase the sentence. It's patently only there for the meter, and as a trick went out of the book 200 years ago. Also, 'verdant grass of Hope' is too flowery. No one writes like that anymore.
Now stare the dry and cracked weeds of Doom. ...how can weeds be 'cracked'? you've lost control of the metaphor. You jumped from dry to cracking, which is a natural connection, but doesn't apply in this case. 'Crackled' is fine, but doesn't fit the meter.
The cruel winds of Age now roam the fields ...more metaphor
Where once Youth’s zephyr played his gentle tunes. ...these metaphors are all over the place. What is a 'wind of age'? Is it stronger, weaker, drier, or wetter than a zephyr? 'Wind of age' does not bring any picture to my mind - it is an abstract, meaningless concept. To draw a contrast, you can talk about the winds of youth as being west winds (zephyr), and those of age as being north winds (boreas), and the metaphor would still make sense. Secondly, you are already dealing in one level of abstraction by having Troy's past personified as a wind. You are then taking it one step further and having the wind being personified by a youth playing a tune. Two levels of abstraction don't work.
These courtyards, once so full of wine and song
Now lie forlorn, their walls bereft of hope, ...more metaphor
And dust from long-lost battles settles still
Upon the fallen ones we loved so well. ...problem with the POV - is this in the present, or just after the war? if the present, 'loved so well' is confusing. If you insist that you love Andromache, then at least change the 'loved' to 'love' or 'have loved'.
Yet through the dark and perilous corridors
Our children stride, their youthful heads held high, ...now it's clear that the speaker is in the past
And bear our faded hopes and dreams with them 
To brave new worlds beyond the distant stars. ... Star Trek?  or did you mean 'distant seas'?