You don't get the highest GDP in 10 years.
You starting point is ridiculously poor compared to the US (the current GDP leader).
There are only a few ways to grow an economy. Increased trade via manufacturing growth, financial strength, technology goods, sales of services etc.; all depend upon either human capital, physical capital (commodities, etc) and/or financial capital.
You have a tremendously poor starting point for all possible forms of capital. 200,000 acres is 312.5 sq miles or 809.4 sq. km. New York City is 1214 sq km. So you cannot possibly have the iron, coal, copper, aluminum, oil, etc. needed to support a multi-trillion dollar economy. Even the resources you have will be hard to develop to large capacity in 10 years since you don't have any significant amount of other capital.
Your starting financial capital is incredibly impoverished. You can't even afford to build basic sanitation or food and water distribution systems. What little population you have will start to die off.
Your starting industrial base is completely non-existent.
You might as well be corrupt, steal what you can and make your escape to an existing country.
Well, that edit the changed the question considerably. As someone already mentioned, Singapore is truly an excellent of rapid growth since 1965 when it became independent of the UK. In 1965, Singapore had a per capita GDP of 500 dollars, not 1 dollar and it already had very nice shipping facilities and a decently educated English speaking workforce
Singapore is also semi-tropical instead of the tundra climate that I would expect in your new island. Compare with Port-aux-Français in the Indian ocean at almost the same latitude. Subsistence farming will be very difficult, might I suggest taking up fishing.
You still don't have the capital needed to build anything significant, so you need to chop down as much forest as needed to built shelters immediately as your people will quickly freeze to death. So, initially it is a race to see whether you die of exposure, or something else.
Assuming you can survive the first year, about the only thing you can really expect to export is food from ocean fishing. As your people are largely uneducated and you have almost no financial capital there is not really a lot more your can do within 10 years by yourself.
You might be able to get some decent fishing boats to increase the harvest and perhaps you could tempt in some human capital to start the basic education processes needed to bootstrap the next generation.
Keep expanding the fishing industry and invite some other countries to come and exploit any offshore oil than may exist. Likewise send out invitations for mineral companies to come looking for gold, diamonds, etc. that may be onshore. Likewise, you can sell fishing licenses for use within your territorial waters.
In short, unless the oil or mineral right bring in lots of cash, after 10 years your people are still dirt poor and will likely remain that way for the rest of natural life of the worlds newest dictator.
The lousy climate will severely limit tourism; average high during the warmest month of the year is going to be about 11.5 C / 53 F. You need money to build 5-star resorts too. I also see gambling and tax havens mentioned -- these bring in both money and corruption. But you still need better climate for real success at gambling, as well as an electric and telecommunications grid for financial services -- again, outside investment again being the key.
It is also true that you will get the short end of the deal on outside investment, unless they are truly dependent on using resources under your control, you will have a poor bargaining position. Your 10 year window is a big problem too. People don't have a reason to trust that you won't confiscate their offshore dollars in your casinos or banks. The Cayman's add 1.2 billion of their GDP to their economy based on 1.5 trillion in financial deposits. Any deal you can strike within a few years will not be nearly as lucrative as the Cayman's have developed over the last 4 decades. Gambling is not the cash cow it used to be either -- too much competition among the casinos these days, margins are down and investors are skittish.
In short, unless oil, etc bring in lots of cash, after 10 years your people are still dirt poor and will likely remain that way for the rest of natural life of the worlds newest dictator.