Not exactly, it took some time before he became crazy. After firing the bullet that contained the Phoenix Force, Wolverine was ripped apart at a subatomic level by the Phoenix Force and died, but when Spidey resurrected him, the Phoenix force stayed inside of him and as time went on, it grew:
We don't know how much of the Phoenix Force Logan possessed during this story but it wasn't just hosting the Phoenix Force that turned him "crazy", it was a consequence of dying, being reborn, hosting the phoenix force, travelling through time and entering in a berserker rage against Czar.
Mojo hired a man named Czar who was armed with the time diamonds (diamonds that give the user control over time and space) and most especially the "Time Bat " (a bat covered with those time diamonds) to mess with Wolverine and Spider-man in order to boost the audiences of his tv show.
Using the time diamonds, Czar sent both Logan and Peter back in time to the Mesozoic era where they lived here for a while. Peter became and hermit trying to find a way to go back to their time while Logan became the leader of a group of primitive people the "small folks" to whom he taught how to make beer, how to fight against other tribes who wanted to kill them and how to be more than just animals. But the Cretaceous-Tertiary extinction event happened and unknown to both Logan and Peter the asteroid that was going to destroy everything was covered in time diamonds and once it crashed on earth it transported them to the future.
Thanks to Logan's actions in the past, the Small Folks, who had been living in the shadows for years, took over after the fall of human society and tried to rebuild the world. In this new world, Peter was a teacher and Logan an hermit seen as a Messiah by the small folks because of his actions during the Mesozoic era. Unfortunately for the earth, something was coming to destroy it: "Doom the living planet" (Doom who transfered his consciousness inside Ego) which prompted the descendants of the small folks to implore Wolverine to help them.
One of the small folks discovered a bunker inside the ruins of a place that belonged to the X-Men, inside that bunker they found the phoenix gun armed with a phoenix bullet containing the power of the Phoenix Force, using this gun would kill Doom but it would also kill its wielder. Peter wanted to sacrifice himself but Logan knocked him out and went to face Doom, he used the gun, killed Doom and was ripped apart at a subatomic level by the Phoenix Force.
Logan stayed dead for a while until, Peter was able to make his cosmic cube worked (prior to Doom's arrival, Peter had been working on this cosmic cube in the hope of using it against Doom but he never managed to make it work). He used it to resurrect Logan and shortly after both of them were transported to the past (Logan back to when Spider-Man just got his powers and was a wrestler and Peter back to when Wolverine was on the run from Dog Logan just after the first Origin mini).
As the Phoenix Force inside Wolverine was growing, Logan started to use it unconsciously as he was travelling through time. First he foresaw the upcoming events that would changed Peter into becoming the hero we know (not stopping the burglar and the death of Uncle Ben), then after being transported in time, both him and Peter were freezed in time by Czar until Logan was able to create a burst of energy with the Phoenix Force that overpowered Czar and restored time.
Next he started to see both the past and the future thanks to the Phoenix Force when he told the history of the time diamonds to Peter and finally as he was fighting against Czar again he entered into a berserker rage and he slowly turned into the Dark Phoenix as a result of everything that happened to him during this story.
Under the control of the Phoenix Force, Logan told to Peter that the Phoenix was hungry and that worlds must burn (similar to what happened at the start of AVX) but before he could dod anything Spider-Man talked him down, Logan regained his senses and was able to reject the Phoenix Force. It's funny but Spider-Man was also really good at talking both the Phoenix powered Magik and Colossus down during AVX when you think about it lol.
After that, they were stuck in time again in the 19th century but this time for more than 3 years until they managed to find a way home.
It's been a while since I read this story but I think that should cover it especially in regards to the context of Wolverine having the Phoenix force.