Pula to Trieste by Bus - Immigration check?

The entry into the Schengen area is at the border from Croatia to Slovenia, so here you will get a Slovene Schengen entry stamp, and sometimes a Croatian departure stamp. Passengers on buses are usually checked pretty thoroughly at this border, and anyone with a non-EU passport is supposed to get a Schengen entry stamp regardless. There are no border controls between Slovenia and Italy, so you will neither get nor need any kind of stamp here.

Immigration at the airport in Rome will check when you entered the Schengen area to make sure you did not overstay, and they should find the entry to Slovenia stamp for this purpose.

You are supposed to get stamps in this case, when crossing from Croatia into Slovenia. In my experience the procedures at these land borders have been somewhat relaxed, but I usually cross by car rather than by bus and I haven't crossed it very much since Croatia joined the EU. I was also crossing without a visa. In your case, with a visa and on a bus, I would expect them to give you a stamp as a matter of course, as they should.

(You will certainly not receive a stamp at the border between Slovenia and Italy because both countries belong to the Schengen area. There are no systematic controls at that border, therefore. Even if you are checked there for some reason, you won't get a Schengen entry stamp because you won't be entering the Schengen area there.)

Your experience in Rome could indeed be problematic if you don't have an entry stamp. If the Slovenian officers don't give you one, you should ask for one.