Did you purchase this sod and install it your self or did you hire a landscaper to do that?
What was the soil like before this sod was laid?

I live in Florida, and soil here is sandy, but I tested pH - it was 6-6.5 -7 in different zones, never higher that 7, and I know that Zoysia can tolerate pretty broad range of pHs, and thus I hoped it would be OK for the grass. Upper inch of my property is not sandy, but dark from previous grasses, I guess, and I added 1 more inch of topsoil before sodding. Yes, we sodded ourselves.

The term "topsoil" is meaningless because all that "topsoil" is is the top 4 to 6 inches of soil from someplace.
Sandy soils, in Florida or anywhere else, tend to drain really well and they do not retain much soil moisture so unless the soil was amended well with organic matter, which will aid in moisture retention, your grass may not have enough water to grow.
To root well grasses need 4 to 6 inches of a good soil, well endowed with organic matter, that is evenly moist but well drained.

What would you advise? Should I spread compost now and in fall, would it help if I add compost little by little every month? Thank you.

You could add, and I would, some compost or other forms of organic matter every month as long as it is not thick enough to smother the grass. Compost probably would be somewhat better initially because that has the wee buggers that make up the Soil Food Web in it so your soil will then have some to later digest other forms of organic matter.

Thank you for your advice: I added peat moss and compost last weekend. I will add them every month. Would you advise to add TurfPro as well - it is much easier to addd than moss and compost? Should I fertilize or compost is sufficient? What else organic matter can I add: I am so desperate to help those grass to survive :)? Should I spray nematodes? What about molasses over compost? would it feed benificial life in compost? Thank you very much!

I have had my Empire Zoysia for three months now with no issues(yet). I just put coffee grinds and soybean meal down.
I'm about to get some cornmeal since it's starting to rain a lot now.