Maybe keep a Norse colony by sending there refugees from Greenland or make the Vinland/Anses aux Meadows survive; alternatively, send there Irish monks.
even if there was a one way trip to the Americas but there happened to be someone on board with the flu or something similar, would this infect the native americans and give them more resistance to later diseases once the europeans rediscovered the continent and possibly give the Amerindians a larger population than OTL and maybe more resistance to the europeans? or nah?
Consider the new crops. With horses and European livestock to trade, the Native Americans might be very interested. Two-way contact does not mean colonization at the levels of the Spanish Inquisition. If the Romans knew there something out there, they would likely build bigger ships. The trade would do much to bolster the craftsmanship and technology of both the Romans and Americans. In many ways, the Americans might have more to gain. If you get up the Great Lakes, you have iron ore. Consider the possibilities.Personally I'd be surprised if at least one ship in the ancient world didn't make it to the Americas and make it back again. But other than an interesting story for the crew when they got home it wouldn't have been a very successful trip and certainly wouldn't have inspired follow up trips. Europe and Africa have plenty of empty land and resources to exploit. In the ancient world there was still plenty of gold left to mine in Europe and trade to India was still open.
What would ancient sailors found in the Americas? Lots of forest and farmers in small villages without any metal tools. Just about the only things the Native Americans have to trade are furs and lumber, which are more readily available in Europe. Even if the Americas were common knowledge there'd be no reason to go. To get sustained contact with the Americas in the ancient world you need to have it make economic sense for them to do so.
Let’s assume the emperor sends an exploration party and when it returns, concludes there are no resources in the New World worth the effort. Assume the journeys result in negligible cultural exchange. You will still have, more than 1000 years before Columbus, a documented quasi-Viking route path to the New World. No explorer in future centuries will be sailing blind; land is known and with natives.
You have a discovery that is documented with the names and accomplishments of the Caesars, their battles, campaigns, etc. A single mission will not return the discovery of continents. It will bring back the latitude of a land populated by Asian-looking people. The latitude would be too far north for India and the speculation between island and continent would not be resolved because the resources to continue such long sailing missions were not there. Scholars will know that if you launch from Britain and sail west, you will reach land. If the launch point is Britain (or Normandy) the information might be preserved there, too.The Americas are very likely forgotten about by everyone except scholars, cartographers, and the occasional sailor telling stories until the at least the 1300s. That's the point at which sailing technology really starts to make traveling across the Atlantic in any regular sort of fashion feasible. Whichever country rediscovers America will have a ball claiming that they've recovered Rome's ancient lost provinces.
There is also a desire for discovery for its own sake. Rome had a wealthy ruling class and if they had reports of a land across the sea inhabited by Asian-looking people and animals they never saw before, there would be some interest.
Eratosthenes (276-194 BC) said the earth is a sphere and the funding would likely be there to investigate.
The real bounty of the New World is not tobacco, coffee and cocoa; but corn, squash and potatoes. Of course, explorers would need to reach Mexico and South America for some of these staple crops.
It’s the same spirit that sent Lewis and Clark up the Missouri River and 1804.