First, the word translated into English (in the NIV) in v. 1 as “advantage” is τὸ περισσὸν, an adjective, functioning as a substantive, declined from περισσός. With respect to people, περισσός is often used in the sense of “extraordinary, remarkable.” LSJ notes,1

As a substantive, then, the neuter substantive τὸ περισσὸν in the phrase «Τί οὖν τὸ περισσὸν τοῦ Ἰουδαίου» means “the extraordinary thing” or “the remarkable thing.” That is, “What is the extraordinary thing about (of) the Jew?” The second phrase is basically, “Or, what is the advantage of circumcision?” Paul responds, “Much in every way!” although it appears he only mentions the first of many things, we must assume, he intended to discuss. No matter, he says that one extraordinary thing about the Jew is that they were entrusted with the oracles of God. He admits, indeed, there was an advantage to being circumcised.
In v. 9, the word translated into English as “advantage” is not the same word (nor a declension of it) that occurs in v. 1. Rather, it is a verb, προεχόμεθα, conjugated in the middle voice from the lemma προέχω. Biblical commentators have offered diverse interpretations of this verse on account of this verb. If the verb were conjugated in the active voice, it would properly be understood in the sense of “to have an advantage; to be better than (someone),” which is actually how the majority (if not all) English translations translate it. However, the verb is in the middle voice, and even Christian Gottlob Wilke (translated by Joseph Henry Thayer) admits the following:2

Wilke basically states that the middle voice in Rom. 3:9 has the same meaning as the active voice although προέχω is never found in the middle voice with that meaning. He argues that the context demands the meaning of the active voice, and so many commentators who interpret it that way (as though it were conjugated in the active voice) can’t be wrong. Not a strong argument, in my opinion. What if we translate it with the meaning it has in the middle voice? According to LSJ,3

If this is the meaning—and grammatically, there’s no reason it can’t be—then it basically means, “Do we put forth a pretext?” or “Do we offer an excuse?” Heinrich Meyer (translated by Moore & Johnson) interprets this clause as such,4

The answer to the apostle Paul’s own rhetorical question? “By no means!” (“Absolutely not!”). Winer (translated by Moulton) remarked,5

The reason that Paul and his fellow Jews could offer no pretext or excuse to avoid God’s punitive judgment: “We have previously charged both Jews and Greeks that they are all under sin.”
In summary, yes, the Jew did have an advantage because they were entrusted with God’s word. However, the Jew cannot offer a pretext or excuse to avoid God’s judgment, for both Jew and Gentile alike are all guilty under sin.
Footnotes
1 LSJ, p. 1387, περισσός, 2.
2 Thayer (translating Wilke), p. 539
3 LSJ, p. 1479, προέχω, 2.
4 Moore; Johnson (translating Meyer), p. 120
5 Moulton (translating Winer), p. 694
References
Meyer, Heinrich August Wilhelm. Critical and Exegetical Handbook to the Epistle to the Romans. Trans. Moore, John C.; Johnson, Edwin. Ed. Dickson, William P. New York: Funk, 1884.
Wilke, Christian Gottlob. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Trans. Thayer, Joseph Henry. Ed. Grimm, Carl Ludwig Wilibald. Rev. ed. New York: American Book, 1889.
Winer, George Benedikt. A Treatise on the Grammar of New Testament Greek. 3rd ed. Trans. Moulton, William Fiddian. Edinburgh: Clark, 1882.