For his own part, Zack wasn't nearly as antsy as Cloud. It wasn't in his nature, and he wasn't exactly the one greeting a person long thought dead. In fact, Zack had no idea that he'd died, or would die, at all. Granted, he'd known that the odds were poor. One SOLDIER against that many Shinra grunts? The chances of him walking away from that fight had been astronomical. The chances of him even being able to save Cloud, still largely catatonic from Mako poisoning, had been just as bad.
Zack could hardly be blamed, however, for thinking that being brought to New York meant that he didn't die. How was he supposed to know any better until someone told him? Sure, he was a little perplexed that Cloud had recovered, and not just recovered, but regained enough function to run his own business, but that didn't translate into figuring out that Cloud was years ahead of him, and he himself had not survived the battle on that cliff outside of Midgar.
Dressed in his old SOLDIER uniform (he didn't have anything else), minus the belt and sword harness since he'd learned not to haul the Buster Sword everywhere he went in New York, Zack strolled through the doors of the Welcome Center right around noon and grinned when he saw his friend.
"Spike!" He launched himself at the kid for an exuberant hug, making sure to mess up that wild, chocobo-style hair with one hand when he did.