This sentence is ambiguous. It can mean "having a difficult time" as fdmaxey suggested and "having a great time" as skye_dino commented. One classic ambiguous sentence from linguistics is "The Queen threw a ball." What does it mean?

It means that the person had difficulty finding his or her way around New York. It is often used when searching for something. "We had quite a time trying to find your house. We took a wrong turn and wound up in the next town."

/quite a time/ should be used when the context gives other hints allowing to figure out whether it was an exciting or troubled time; here you can presume it was a difficult or long time, as the clue is /to find my way/, pointing to a difficulty. Of course in talking the hints come from tone and facial expression, adding meaning. The transcript of a sentence from conversation sometime is ambiguous, while the original spoken sentence was not. conver

The term "quite a time" can mean several things. It hints at a meaning that is implied (assumed to be undestood) as opposed to explicit (spoken directly). Used in the example sentence you gave, it probably means "quite a (difficult) time." Now, if someone says "we had quite a time last evening," it probably means "we had quite a (good) time."

you can understand that, here quite a time means spending a reasonable time around or in newyork.and knowing about city