Upvote:4
The last Trenitalia train from Ventimiglia to Imperia departs at 21:20. To catch that train, you would have to get the TER from Nice departing at 20:28 (arriving at Ventimiglia at 21:13). This is probably implausible given a 20:00 flight arrival, particularly if you will have to deal with passport control, customs, and/or baggage claim upon your arrival in Nice.
If money is no object, and you can't find a rental car, Rome2Rio suggests hiring a towncar at a price of about 360€. Otherwise, you may be better off getting a hotel in either Nice or Ventimiglia and completing your journey by train on Christmas morning.