> I'd try to invent ... a printing press
I sometimes wonder if the development of the printing press relied on technology that hadn't been available previously - like many/most innovations. But what?
Paper? There were ink-retaining sheets long before the printing press. A durable mechanism for the roller? They made wagon axles, I assume, that supported much more weight. Durable letters? Even sans metal, I'd guess that carving wood letters would still be worth the effort.
Gutenberg's press was a modified wine press that was based on a screw. The Romans had screw wine presses. The precursors were available to the Romans.
He used lead letters. Lead was readily available in Roman times - after all, the water pipes were made of lead (and poisoned the people who used them).
The lead letters would quickly wear out, but it was easy to melt and recast them as needed. I've seen a demonstration of it.
I think the Chinese did wood block printing, but it didn't get very far.