Well yes and now - the ancient Polynesians colonized all of the Pacific over vaster distances. The key thing is motivation: What motivated the European age of exploration? Note that it started in the Iberian peninsula and was boosted after 1453:
Why would the Romans want to improve sea trade when they controlled the land trade? As it is Rome was losing gold to India for purchasing spices and cotton. There was no point
And even more so, an open sea blue water sailing. No point
Even more so for pre-Assyrian Egypt - they had need of next to nothing: maybe some lapis lazuli from Afghanistan, but beyond that? meh.
The monsoon trade with India antedates Roman expansion into Egypt, apparently dating back to Sumerian times. The Sumerians themselves wrote that they arrived in the region with their oldest presence being in Persian Gulf ports, and their placenames (rivers, cities) are not Sumerian words.
Egyptians themselves had an intermittent but continual interest in expanding beyond the Nile valley, into Sinai and beyond, even in the predynastic or "dynasty zero" era. Egyptian navigation of the Red Sea reaches back at least into Middle Kingdom times.
'The Story of Wenamun' has an uncertain date but records (or dramatizes) a Late Kingdom voyage to get cedars from Lebanon, a practice that began well before the Late Kingdom, as there are Old Kingdom examples, but those could have arrived via merchant ships from what is now Lebanon.
Most of the Punt keyword, sorted: