We have no solid information about the time when the East Road was constructed, but the fact that it ran between the Grey Havens and Rivendell implies that the route was originally established in some form by the Elves, presumably at the time of Rivendell's founding (in II 1697) or not long thereafter.

After the establishment of the Kingdoms of the Dúnedain, numerous building projects were undertaken across Middle-earth, and we know that the great North-South Road was established at this time. The forces of the Last Alliance mustered at Rivendell, and it's established that the Dúnedain built bridges to help the movement of their troops in this period, so it seems plausible that the East Road may also have been reinforced and improved during the closing years of the Second Age.

Whatever the details of its first building, the road was three thousand years old or more at the time of the War of the Ring, though doubtless it had been repaired and maintained during that time, especially during the first two millennia of the Third Age when most of its length still fell within one of the realms of the Northern Dúnedain.

We know that the road extended westwards to the Grey Havens, but we have no detailed map showing the final stages of its route. Given the importance of the Tower Hills to the Elves, it seems likely that the road would have passed close by those hills, or perhaps a lesser roadway would have run southward from the main East Road to Elostirion and the other Elf-towers.