We will be centered in Zagreb and go to the Istrian Peninsula, Rovinj, some towns around Zagreb and a day in Slovenia. How is the weather there then and are a lot of places closed?
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Slovenia
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Inland, early April is early spring, the temperatures will begin to rise, trees just starting to grow their leaves again, etc. Expect at least some rain, the weather is very changeable and unpredictable during that time, daily highs will generally be in the 50s and 60s range, lower inland, higher on the coast, where the weather will also be somewhat better overall, with spring in full swing.
Of all the places you mention, only Rovinj will be a bit sleepy, the others will be operating as usual. However, if you happen to visit Rovinj on a sunny weekend, I'm sure you'll find there will be quite a few visitors from inland there, as well - very few tourists, though.
I do recommend more than just a day in Slovenia, though, there are many places well worth seeing there. Spend a night or two at Lake Bled or in Ljubljana, otherwise you'll end up spending a large chunk of your day trip in the car.