I understand that the IL-62s are due to leave the fleet, as are the AN-24s. If they are still operational it is probable that they are just there for back-up purposes.
However, in theory, Cubana operates its IL-96s on routes from Cuba to Buenos Aires, London Gatwick, Madrid and Paris-Orly. As TSR2 rightly mentions, they did have a few problems earlier this year and had to charter in a 767 to cover flights for a while; but I believe IL-96 equipment is now back in use. Cubana now also has TU-204s in operation, both as passenger and freighter aircraft. You can - again, in theory - catch a TU-204 on the routes to Bogota, Santo Domingo and Caracas. Finally you can usually still catch a Yak-42 on the Havana-Cancun route daily.
I'm afraid that opportunities to fly on an IL-62 are getting fewer and farther between. You might be lucky with Air Koryo on the Beijing-Pyongyang route, though they now use the TU-204 too.