So there's a few things here. First, the Phoenicians could more or less be reliably counted upon to fight the Carthaginians-there's no reason to expect that they would disobey Alexander, given his reputation-they need to look no further than what happened to Tyre, a city who's transgression was agreeing to everything short of allowing his army enter the city, to know that defying Alexander was not a realistic option. Second, for the same reasons, there shouldn't be any significant rebellions in Alexander's realm while he's still alive. It's after his death where the fun starts.
Third, the campaign itself. The most likely route to Carthage is through Magna Graecia, allowing for him to also subjugate the Greek city states there under his fold. Doing so at least nominally won't be that hard, as the minor city states don't really have the will or the means to resist, and those like Tarentum and Syracuse will wager its better to let him be on his way and bide their time for dealing with Macedonian control at a later date, preferably when his body is cold. Syracuse in particular might be very interested in an alliance with Alexander-they could make out handsomely seeing their one rival on the peninsula destroyed.
Anyway, I don't see why the initial campaign in Sicily wouldn't be any less successful than Pyrrhus's was up until he reached the walls of Lillybaeum. This is where the first major test would come. Taking Lillybaeum almost assuredly requires cutting it off from the sea, which means taking on the Carthaginian navy. While this is a fight Alexander should win, Carthage can be expected to put up a strong fight. Assuming victory here, Alexander's next immediate objective would be Carthage itself, setting the stage for another one of his epic sieges. Once Carthage is taken, this does not necessarily mean Alexander inherits Carthage's connections and hegemony over the other Phoenician cities dotting the western Mediterranean-this will itself require a some investment of time, using a mix of diplomacy and force. It would be interesting to see how Alexander would handle a more tactful and patient approach to dealing with foes/wary neutrals than straight military intervention.