In my opinion this is a (at least) two steps process:
- Make the fruit's peel more durable.
- Coating the inner part of the object.
Make the fruit's peel more durable.
The conservation of fruit peels (seems strange but is true) is a well know problem in the field of restoration of contemporary art.
Zoe Leonard "Strange Fruit (for David)" is the classical case of study about this topic.
Zoe Leonard after the death of a friends of hers begun to sews the peels and do with different material to recreate the original shape of the fruit. This work of course was very degradable so a German conservator Christian Scheidemann developed a method to treat the fruit peel to make them more durable. He used the Paraloid B72, a well know ethyl-methacrylate copolymer used in conservation of cultural heritage, under vacuum. I'm quoting from the link above:
Early on, her dealer, Paula Cooper, suggested the possibility of preservative intervention for the sculpture. Leonard was amenable and worked for two years with German conservator Christian Scheidemann to devise a way to arrest the decay of the fruit surfaces. After much testing, Scheidemann developed a solution that consisted of shock-freezing the pieces and then penetrating them with Paraloid B72 under vacuum. This solution was complicated by the need to protect the wires, threads, and other elements from the Paraloid B72; in other words, the piece presented the intricacies typical of any mixed-media work. But Scheidemann succeeded in this as well.
A generic more detailed research about food conservation can be found in this here.
Coating the inner part of the object
As other user stated generally plastic glasses and other for drinking are made of polystyrene, PET or polycarbonate. The third one is more durable and use for stuff used more times. However coat a cup or a vase with plastic on you own is quite dangerous (there are chances of poisoning if done in the wrong way) and expensive and difficult.
If you don't use vacuum you should use blow molding over a mold of the interior of the fruit (so you have to make a first mold and then another "negative" mold) and the attach the treated peel in a second moment with some adhesive, very expensive process. I could go to a Murano glassmaker with your mold! In this case you can add a plastic or glass edge to your cup I don't think is very nice to drink from a plastic peel.
There is an alternative cheapest way, using Food Grade Silicone Mold Rubber this is probably what I would used you can find in internet a lot of types of silicone rubber that can fit your needs.
However I don't think you can later wash it in the dishwasher!
As stated Jaroslav Kotowski you can use even a more expensive epoxy resin however you should pay attention to use a food grade epoxy resin. Non food grade epoxy resin could be very toxic.