I'm not sure how much you'll find open in Ljubljana on Sunday. That's the day I was there and the large shops were closed. The churches will close in the afternoon. The restaurants and cafes were open and people were walking around. There was a craft market near Dragon Bridge and the nice and shady pedestrian area with cafes was busy.

You might try this fellow. He's Croatian but he mentions Slovenia in his post so he might have some information or help for you.

http://www.fodors.com/community/europe/im-croatian-ask-me-anything.cfm

You better not think of using public transportation, if you have such a small time window. You probably come on shore at 08.00 and have to be on cruise till 17.00, correct? Traveling time with train takes 2 hours (ticket is arount 10 € per person), with public bus it takes at least 1h45m and ticket costs also around 10 €. One way of course.
Visiting Ljubljana is not a problem. All major shoping centers are opened till 15.00, restaurants and cafes are opened till 22.00 at least. On your visit you can see old part of Ljubljana, with nice architecture and Ljubljana's castle as well.
So, you might want to consider to take some private transfer company, especialy, if there is more of you. You might want to try ask with After Fly transfers company (www.transfer.si) and you might get some guide there as well. If you don't see visiting Ljubljana that interesting any more, they might have some good suggetions about spending your time on shore.
But, if you decide to stay on Slovenian coast, Piran is best place you would want to visit.

Thanks for the advice. We arrive in Koper at 8:00 and depart at 18.00. I had read some nice articles about Ljubljana but four hours of just travelling to get there and back is probably a bit much, especially on Sunday when things will be closed. We'll have to leave that for a future land trip.

It will be our first time in Slovenia so we would just like to have a nice day's itinerary. From what I've read Koper doesn't have much to see so we're looking for more interesting places to visit. Information on Piran and how to get there would be appreciated.

We will also be going to other ports on the coast - Split and Kotor which I'm really looking forward to, and Dubrovnik where we have been once before.

We were in Dubrovnik a few years ago in August and it was increibly hot and packed with tourists. We did do the walk on the wall and I don't need to do that again. So, any other suggestions? We could take a trip out of town if anyone has some good ideas.

Thanks for your help!
Linda

How about Trieste? I haven't been there but the old town looks charming. Do a search here - I remember someone posted about going there within the last year.

It's a larger city and more popular and only a half hour from where your ship docks so there's apt to be public transportation.

I have just been to Ljubljana, Piran and Trieste a few weeks ago. Ljubljana and Piran are wonderful, Trieste is a big city, if you enjoy big Italian cities it certainly has some impressive architecture and I am glad I stopped there for a day (between Piran and Verona) but I am not sure everyone would think it worthwhile.

I really loved Ljubljana - stores being closed would not matter to me, it's the city you want to see I assume, not just to go shopping (you can do that anywhere). Plenty of restaurants are open on Sundays. The problem would be getting there as buses in Slovenia are slow and not all that frequent. If you really wanted to see it I would assume you could take a taxi.

Piran is closer but not sure the bus schedule would work.. Google it, don't recall the website but all the schedules were on line. However, it's so close that a taxi should be quite reasonable. I took a taxi from Piran to Trieste for 50 euro because the buses were not going when I wanted to go. I never take taxis, anywhere, but in this case it was worth it. There seem to be lots of drivers around willing to take people between cities. Kopor is between Trieste and Piran so it should be reasonable to go to either one.

Piran is a beautiful small town with great main square, church on a hill, town walls, well worth several hours (I spent two days).

I'll be posting a trip report and photos when I get home but not sure if it will be soon enough to help you. I am currently still over here but going home in about a week. If you have more questions about any of those places I'll try to answer them.