Holland answered the problem of land with dikes, levees and the application of a new technology - new way back - wind power. Most, not all islands have barrier reefs which can be extended upward making a levee system, then it is a process of pumping out water. Of course in the age of melting ice and ocean rise your island may not be an island for long
Desalting ocean water is difficult, even with modern methods - There are other ways to secure fresh water and power.
http://www.wired.com/wired/archive/13.06/craven.html has a tried and true method.
Distance from the rest of the world is not a problem. Humanity has found ways to travel all of the world rapidly and with ease, in the form of aircraft and ships.
If the energy resources are too much for you the blimp and zeppelin are great methods - with today's materials and knowledge we could build them far, far safer than the Hindenburg filled with hydrogen and covered in flammable cloth. Perhaps that could be one of the industries developed and sold from the nation? An export item?
Or if you do prefer ship travel wind power is still there. We gave it up for the "faster" steam, however we have come along way since the days of the cloth sail, there are other options such as fixed wing designs:
http://www.wingsails.com/cetiri.html The only reason why these are not being applied to larger cargo type craft is because mankind is lazy or because we are an oil based economy that doesn't care to invest the time and energy? There are a lot of different designs out there, a few innovators who due to lack of funds and no market closed up shop. A few years ago a man designed a sail boat which used an aluminum wing and was piloted from the cockpit with throttle controls and steering wheel. He closed shop because he couldn't sell the boat.
He couldn't sell the boat because when it comes to sail power people want whipping sail cloth and are trying to harken back to the day when... Not because they are looking at an economical and modern approach to transport across the water.
Need energy? We all know of solar and wind power - again I point out
http://www.wired.com/wired/archive/13.06/craven.html same basic technology this time converting the energy into electricity.
How tall can a building be built? Now days soil and geology does not stand in the way of impressive structure being built. We know how to sink pylons deep down to stabilize the foundation. There are some very tall structures build on reclaimed land, sand, and other geologically unstable materials that start off with deep pylons which reach down to bed rock.
Miami is a good example, all of those tall hotels do not set on stone, they are set on stand with pylon supports making it possible for them to sit and not sink into the sand/coral of Florida as close to the beach as possible. Resorts, got to love them.
Food. If you are thinking of crop lands that stretch out into the horizon, using lots of chemical fertilizers, pesticides and requiring irrigation either through canal or pipes then yes, it is a problem.
But we are kind of smart, as apes go. We choose the old methods because it it cheap and there is land available, however there are methods that can be instituted where vertical acreage can be had. Hydroponics comes to mind, many organic successive planting methods are also intensive growth methods, meaning each plant is place closer together maximizing the soil area. If you think about it lettuce only needs two feet of vertical space - if you tray and stack or "shelf" lettuce you can on the same amount of land area triple or quadruple the land area. This can be done either through artificial lighting, or through narrow "shelves, which maximize angled light. Since the sun moves across the sky spending little time directly over head your plant can get its required 8 hours of direct sunlight. BTW many vegetable plants have small root systems, the requirement of land us usually for the top growth (leaves and fruit) and for the ease of presently use machines to cultivate and harvest the field. In which I add that those machines are so 20th century - we can do better.
Considering that most islands will not have a natural supply of fresh water, then you are going to have to irrigate the crops anyways - might as well move the water up as well as out.
Further, and Island is surrounded by water, in that water is fish and algae, and kelps - plenty of people eat these sorts of foods on a near daily basis. It isn't too difficult to farm the ocean surface either.
Any other problems? I'm willing to bet that any problem you think of there is already a solution, maybe not being used due to present economic systems in place, but a solution nonetheless.