My interpretation of the lyrics are a combination of ideas that play on zodiac signs, film noir/vintage espionage crime drama, and being seduced by a dangerous women, where the main character is blind-sided by a women, and in which he usually doesn't fall for just one woman ("you've thrown away your choice").
Well morning comes and you're still with her And the bus and the
tourists are gone And "you've thrown away your choice you've lost your
ticket So you have to stay on But the drum-beat strains of the night
remain In the rhythm of the new-born day You know sometime you're
bound to leave her But for now you're going to stay In the year of the
cat
The year of the cat for instance is not a real zodiac sign in either western or Chinese zodiac, but cats themselves are often symbols of power relating to seduction, mystery, and magic and have been culturally worshiped (ie: Ancient Egyptian Goddesses) or feared (ie: Middle Ages Christian beliefs centered on black cats being bad luck).
The Cat is one of the twelve signs of the Vietnamese zodiac. It
corresponds to that of the Rabbit in the Chinese zodiac. At the time
of the song's release, the most recent Year of the Rabbit had been 11
February 1975 to 30 January 1976; thus, the song was recorded in the
Vietnamese Year of the Cat.
However, they are featured in Vietnamese Zodiac where the cat corresponds with the Chinese' Zodiac's Rabbit, which happens to be the year the song was recorded in.
But really, the idea behind the zodiac or really, horoscopes, is that a person is born with certain traits and that they may have a semi-predetermined "destiny". In the Chinese Zodiac people also often look to the zodiac year, hence the man in the song can not fight his urges for this women, because it's 'the year of the cat'. It's a metaphor for the time he was seduced by this women, the women being symbolized with the cat...
But there are lines like,
On a morning from a Bogart movie In a country where they turn back
time You go strolling through the crowd like Peter Lorre Contemplating
a crime
This may be alluding to the film, The Maltese Falcon where actors Lorre and Boggart worked together for the first time. It's a film nior film about a private investigator, a feminine fatal, and a jewel-encrusted falcon statuette.
The Maltese Falcon is a 1941 film noir with screenplay by and directed
by John Huston in his directorial debut, and based on Dashiell
Hammett's 1930 novel of the same name. The film stars
Humphrey Bogart as private investigator Sam Spade and Mary Astor as
his femme fatale client. Gladys George, Peter Lorre, and Sydney
Greenstreet co-star, with Greenstreet appearing in his film debut. The
story follows a San Francisco private detective and his dealings with
three unscrupulous adventurers, all of whom are competing to obtain a
jewel-encrusted falcon statuette.
However, Peter Lorre and Humphrey Boggart also went on to do more iconic romantic adventure film noir, Casablanca, which is more likely what the song is referring to.

Casablanca is a 1942 American romantic drama film directed by Michael
Curtiz based on Murray Burnett and Joan Alison's unproduced stage play
Everybody Comes to Rick's. The film stars Humphrey Bogart, Ingrid
Bergman, and Paul Henreid; it also features Claude Rains, Conrad
Veidt, Sydney Greenstreet, Peter Lorre, and Dooley Wilson. Set during
contemporary World War II, it focuses on an American expatriate who
must choose between his love for a woman and helping her and her
husband, a Czech Resistance leader, escape from the Vichy-controlled
city of Casablanca to continue his fight against the Nazis.
While listening to the song (which I have never heard before, but I really like it, so thank for this), there are something about the instrumentals that are reminiscent to both 70's romance and crime drama films, but also has some throwbacks to something more jazzy, if you will.
The following however is the interpretation the Year of the Cat Wikipedia page states,
Co-written by Peter Wood, "Year of the Cat" is a narrative song
written in the second person whose protagonist, a tourist, is visiting
an exotic market when a mysterious silk-clad woman appears and takes
him away for a gauzy romantic adventure. On waking the next day beside
her, the tourist realises, with equanimity, that his tour bus has left
without him and he has lost his ticket.
So the man in the song is also meant to be a literal tourist and not just a metaphor for being out of his element. This kind of points stronger to Casblanca IMO, as the characters in the film have to present "letters of transit".