By being an island arc volcano.
You seem to be referring to hotspot volcanoes, which are fed by abnormally hot regions of the Earth's mantle. As the continental plates drift by overhead, the area fed by the hotspot changes over time, and so you end up with an island chain in which the older, dormant islands have eroded to almost nothing, but the islands still being fed by the hotspot (and therefore, still volcanically active) are much larger.
Hawaii is a good example of an island chain produced by a moving hotspot. You can clearly see the movement of the hotspot over time, from Kure (formed ~28 mya and now completely eroded, leaving just an atoll) to Hawaii, where it now approximately resides.
Island arcs, however, are chains of volcanoes that form along subduction zones: places where two continental plates meet, and one is forced beneath the other. This feeds volcanic activity, and as long as the plates continue to subduct, this activity will continue.
The Aleutian Islands, at the boundary of the Pacific and North American plates, are an example of an island arc. They are thought to have formed around 50-55 mya, and are still volcanically active. Just as you can trace the path of the Hawaii hotspot by looking at the islands, you can trace the Pacific/North American plate boundary by looking at the Aleutians:
So all you need is for your island to be located at a subduction zone that's been around for 100 million years or more. Tectonics will take care of the rest.