Things to consider
Location
Volcanoes generally appear where tectonic plates diverge and converge(Pacific Ring of Fire). They also form when there is stretching and thinning of the Earth's crust(East African Rift). Lastly there are HotSpots or Mantle Plume volcanoes(Hawaii), which have been hypothesized to to arise from upwelling diapirs with magma from the core–mantle boundary, 3,000 km deep in the Earth. You would not be able to pick a spot at random and demand a volcano appear - picking a location would include a lot of timely and costly studies.
Hazards
Volcanic Gasses, Acid Rain, Lava eruptions, Ash, Earthquakes, Fumaroles, Mud pots, Geysers and Hot Springs are all associated with active volcanoes and can potentially make your island a nightmare rather than a tropical paradise.
Inhabitable?
It takes a long time for dirt and sand and vegetation to develop on a barren rock island. Will you just throw good money after bad to turn your isolated sea rock into a tropical paradise? What are the logistics of this?
Possible Tatics
Nuclear Bomb -
This would need to be delivered deep into a theoretical magma well to work. The idea being that the the explosion in the magma will cause it to continue to erupt. This has not been tried, though volcanoes flows have been bombed on several occasions throughout history (Hawaii x3, Etna x1) to no effect.
Drilling - The idea behind this is that the by doing so, you would release building pressure causing the magma to flow through the hole created. This actually happens occasionally with current active volcanoes to little effect except some unexpected damage to drilling equipment. The drill hole is just too narrow to transmit the force of an eruption. Most mantle drilling boreholes are only 30cm wide.
Water Injection - Magma will also erupt explosively with the addition of outside water, this is what happened with Eyjafjallajökull in 2010. If the right ratio of water-to-magma exists, then the explosive mixing of magma and water can be self-sustaining, meaning that the explosive eruption will continue propagating until the supply of water or magma runs out. However, too little water means that you likely don’t sustain the explosive mixing, too much water and you quench (solidify) too much of the magma.
Practical Application
Think of a volcano as a bottle of champaign. When you pop the cork some less dense liquid is going to rise up until it hits level with the surface - like the lava flows of Kilauea. If you want a properly explosive eruption you want to make bubbles, Lots. You could decompress the magma (forcing gas to leave the solution), you could crystallize minerals to concentrate water and volatiles in the remaining magma or you could heat the magma with a new intrusion. Once you've created bubbles, you need to concentrate them at the top of the magma. Back to our bubbly - think about what happens when you shake that bottle before popping the cork.
what you'll need to do to have a hope of making this work is;
- Find a volcano that is already showing some signs of magma intruding at shallow depths. This might be high levels of volcanic gases, shallow earthquakes, deformation of the volcano. You want something “primed” to go.
- Figure out a way to release the lithostatic pressure keeping the “cork” on the volcano so that the bubbles can form.
- Figure out how to get a lot of water into the volcano quickly … but not too quickly.
Delivering a Nuclear Bomb into the magma well doesn't address the pressure issue or remove enough of the overlaying rock matter to release the lithostatic pressure. Drilling into a volcano is too small - like poking a pin hole in a balloon. Oddly - both of these options are thinking Far too small.
So, Lets think big.
First your multi-billionaire will need to find an volcano that doesn't erupt often(pressure) and has a decent amount of magma to erupt. If you're wanting to "create" an island from scratch, it needs to be underwater, but not far underwater that it takes more than a few years to hit the surface and start forming an island - you wanna enjoy this thing before you're dead. if the magma is already rising under the volcano, then you need to catastrophically release the overlying rock to create bubbles in the magma to concentrate pressure. Lastly we need to introduce water to help exposivity.
So we know what we need to do - but how do we do it?
Lets start with high explosive charges just below the surface of the rock bed to get rid of the land above our magma body. The resulting shaking may even help shake bubbles free in the magma, while more bubbles release due to the release of pressure. If the pressure increases enough the weakened roof may give way above the magma body which would allow decompression. Underwater, this would also allow the introduction of seawater which will get the party started.
Seems like we have a pretty good set up, right? Eh, Kinda. Remember this is all theoretical and no one really knows how long it might take from the trigger of an eruption to get the actual eruption itself. Looking at history, there are no good indications. Mount St Helens (1980) blew seconds after the earthquake and landslide that triggered it. In Chile volcanoes have triggered months to years after the large earthquakes thought to have triggered them. There are many types of triggers which can lead to volcanic eruption and they tend to be unique to each volcano and the geology/geography of the area. furthermore, volcanoes more often than not do not erupt when triggered. Its just really complicated process that we don't really know much about. Its a lot of trouble to go to with no guarantee when or if you'll get the reaction you're hoping so. You might even prevent an eruption rather then speed it along.
Conclusion
Theoretically, Yes - this is a possible scenario. There are a lot of complicated logistics involved, you would need conditions so perfect it's almost down to luck even after the planning but triggering a volcanic eruption could be done. It could not, however, be done in a controlled manner. Once you carry out the triggering actions there would be absolutely no way to guess or control when an eruption would happen, what it would look like afterwards (There are many different types of volcanoes from Fissure Vents to Lava Domes to your classic Volcanic Cone.) or even how long it would continue to erupt.
