Contrary to reports like this one, Muqtada can't really "study to become an ayatollah," at least not in the way the story frames it. He can study to become a mujtahid, which means he is authorized to practice ijtihad, rational examination of the scriptures (coming from the same root as jihad, or struggle, in the sense that ijtihad is an intellectual struggle, the exertion of scholarly effort) and issue decisions, or fatwas, in response to questions posed by adherents.
One achieves the rank of mujtahid through a fairly formalized course of study. To reach the rank of ayatollah, however, requires, in addition, a significant body of published scholarly work, a substantial following who recognize him as a marja al-taqlid ("source of emulation," a guide to correct Islamic practice), and, importantly, recognition and acclaim by other mujtahids, students, and clerics. The title of ayatollah is bestowed on those who have exhibited special insight into scriptures, excellent facility with the Arabic language in its most complex grammar, and superior juristical chops, qualities that Muqtada, by all accounts, does not possess. (Sadly for noted video game enthusiast Muqtada, lighting up the scoreboard in FIFA 06 is not presently one of the categories in which aspirants to the rank of ayatollah are judged, but times do change ...) Even if he is recognized as a mujtahid (as is probable, given that, as Juan Cole notes, many of his instructors were followers of Sadr's father, and also: Do you really want to fail a dude who has his own militia?) I think it's rather unlikely that he will be recognized as an ayatollah, and certainly not any time soon.
But what's really interesting and significant about Sadr is how little this matters. Because of the political power he currently holds in Iraq, and the size and depth of commitment of his movement, Sadr already wields influence to rival an ayatollah's. Up until now, he's had to rely on the fatwas of senior clerics for his formal religious legitimacy. Once he achieves the rank of marja, and is able to issue his own fatwas rather than relying on the credentials of sometimes uncooperative allies, his political power will more than suffice for the influence that would come were he recognized as an ayatollah. It remains to be seen how overt a role he intends to play in Iraqi politics, but there's no doubt it will continue to be a substantial one. Sadr has given conflicting answers about his support for Ayatollah Khomeini's theory of velayat e-faqih, ("rule of the jurist") which, despite being the basis for the Iranian government, is still considered a marginal theory, if not outright heretical innovation, by the majority of the world's Shia scholars. It's more likely that he'll try to establish a system of government in line with the theories of clerical activism developed by his revered uncle, in which there is religious leadership but not outright rule. In addition to having more scholarly acceptance than Khomeinism, these ideas also have the benefit of being Iraqi-grown.