First let's try to understand the language, and not have its meaning obscured by translation. Here is a more modern (and in my opinion generally excellent) translation, by Christopher Rowe:
Because the unjust person is grasping, his sphere of operation will be goods — not all of them, but those to which good and bad fortune relate, ones which are generally good, but not always for this or that particular person. (Human beings pray for these, and go after them; but they should rather pray that what are goods generally speaking be good for them too, while choosing the things that are good for them.)
And Sarah Broadie's commentary on 1129b1-2 (found in the same volume):
grasping [*pleonektês, lit. 'one who goes in for having more']
(a) 'Grasping is always perjorative in Ar. as in Plato, hence cannot be used of the laudable character who seeks more and more of the goods that depend on personal effort alone and are always good for one, namely excellence and excellent activity.
(b) At EE VII.15 (=VII. 3), 1248b25-33, excellence is said to be the condition under which the things that are 'generally good' (there called 'natural goods') are good for the individual. Thus the prayer at line 5 should be for excellence.
Nevertheless, we are left with ambiguity. The problem is that Aristotle is using ἁπλῶς which might mean "generally", "simply", or "absolutely" (and apparently "unconditionally" in your translation). And it isn't guaranteed that it means the same thing in each place! Translators are inclined to use the same English word for the same Greek word, especially within a passage, even if they are conscious that it makes things hard to understand.
Here we have the ἁπλῶς good being spoken of as if there are two kinds of ἁπλῶς good. This may be a deliberate play on words by Aristotle: in 1129a25ff Aristotle is talking about the homonymy of justice, perhaps he is continuing to riff on this theme.
How can something which is ἁπλῶς good be good only for some people?
Something which is "unconditionally good", as your translation puts it, surely can't be good only for some people. So what is going on?
Let's look at the surrounding facts describing the grasping person's "good's" apart from the fact they are ἁπλῶς:
- they are related to good and bad fortune
- they are not always good for this or that particular person
- they are the goods people pray for
Result? "Unconditional" or "absolute" is probably a bad translation. Aristotle seems to have in mind goods that are both dependent on some feature of the person who receives them, and which are up to chance. These aren't qualities that jibe with "unconditional".
When discussing the "unconditional (or generally) good" for the greedy or grasping man, I suspect Aristotle has in mind things like: money, food, power, and sex. The general public speaks of these as unconditional goods, but in fact are conditional. If you're surrounded by robbers, must fast for medical reasons, are set upon by revolutionaries, or get the clap, they aren't unconditionally good. Perhaps more subtly, he is pointing towards the fact that money (for example) might be spoken about as if it is good, but having too much of it might also lead us to do things that both corrupt us morally and lead us to eventual ruin.
My diagnosis of the passage is that the first mention of the ἁπλῶς good is a homonymous use: here an appropriate translation is "general good". The second mention of the ἁπλῶς good is used in the same way Aristotle often speaks, to indicate "good without qualification".
What does Aristotle mean by "good without qualification"?
I'm happy to accept Broadie's commentary that the parenthetical prayer is about the "natural goods" described in the Eudemian Ethics. Basically: (1) we should pray to be excellent, and (2) it's better to pray for, e.g., the wisdom to use money so that it is good for us (i.e. for munificence and openhandedness), than to simply pray for more money.
In this regard, I'm happy to adopt @virmaior's answer insofar as it instructs us how ἁπλῶς goods might be pursued in a defective way.
This understanding of the unqualified good isn't merely based on this specific reference to the Eudemian Ethics. Aristotle's ethics is both philosophy and instruction manual. One of its constant normative themes is that the only good a person should pursue is to perfect their character. This is accomplished by making good choices and surround themselves with similarly virtuous persons.