Hello

. He’s a grand age and looks adorable. It does look a bit swollen round his eye and it may be infected.
I completely understand the worry about stressing them with a vet visit in old age - however sometimes it’s necessary and you can make it as stress free and comfortable for him as possible. He could be having pain and discomfort from that eye so it needs treating. The vet may give antibiotic eye drops or oral antibiotics. It may be his nails have got a bit long as he has got older as well.
If it is infected then untreated it could lead to more unpleasant things.
The advantage of taking him to the vet’s now as well is you can also ask if he can have some Metacam. It’s the equivalent of human ibuprofen and at his age they tend to have toothache, arthritis and all sorts and have lots of aches and pains! They are very good at hiding pain so you can’t always tell.
He is bound to be a bit bothered for a couple of days after a vet visit but just give him lots of treats to distract him. And Metacam could give him a real boost as well.
The main thing is to find an exotic vet - they know more about hamsters- the cost is no different usually they just specialise in exotics within a vet practice as opposed to cats and dogs .
To take him you’d need a guinea pig sized pet carrier half full of bedding and some torn up strips of toilet paper and a little house (cardboard box would do), some food, a piece of cucumber and a couple of treats.
Whether travelling by car or public transport, I hold the pet carrier on my knee (as a passenger in a car) and have a blanket or towel over the top so it’s dark and cosy and blocks out lights and noise etc a bit. I also talk to them in the pet carrier. He may just go to sleep in there.