Question: If you're playing solo with one deck, you can engage Shelob yourself before exploring this location, right? It doesn't say you cannot. So since Shelob is already engaged with you, the "Shelob engages the first player" doesn't kick in. Am I right?

As long as you are in stage 3 of the quest you can engage Shelob as normal. However, bear in mind that she still makes her quest phase attack even if engaged.

Once Cleft of Cirith Ungol is explored, Shelob engages the first player, which means she will follow the the first player marker in the refresh phase.