What economic and development lessons/strategies can developing Caribbean countries learn from successfully small island nations like Singapore?
I like this question, in the sense that although it is broad, I think it can be answered concisely and factually.
Singapore and China (from your previous question) are largely authoritarian countries. So they were both able to exert a lot of government influence over their development programs. In this respect, they are not similar to Caribbean countries.
However, in other respects, Singapore is a good case analog because it is an island with very few natural resources and a relatively small population with correspondingly small domestic economic prospects.
Here are some pillars that underpinned Singapore's meteoric economic development.
Create stable and market-friendly government. Economic development takes a long time, and it's hard to execute it if government changes frequently. Also, governments need to be prepared -- over time -- to relinquish control over their economies as they transition to market-based. Both Singapore and China accomplished this deliberately and in an ordered fashion (moving too quickly can cause chaos...see Russia for example).
Create stable currency and a good banking system. Singapore, like many developing countries, decided to peg its currency to the US dollar, which gave it exchange rate stability and familarity. Banking systems are crucial to ensuring liquidity and investment for economic growth, so installing a currency board, conservative fiscal policy, a decent bank, and a set of quasi-government commercial banks helped create financial infrastructure from scratch.
Invest in general economic infrastructure. Singapore carefully built out physical infrastructure (roads, communications), legal infrastructure (courts and business law), and talent infrastructure (through schools and workforce training) to reduce economic friction, create economic resources, and maximize economic opportunity for development.
Seed strategic industries. Singapore identified major industries that could help drive economic growth, and used a combination of government subsidies, pro-business laws, and economic protectionism to nurture these industries. These investments included: making Singapore a low-cost manufacturing hub and a shipping hub (early economic development), and then subsequently a banking and communications hub (later development).
Assist trade. The Singapore government worked closely with private and semi-public enterprises to help push through trade negotiations with countries (US, Japan, China, etc), reduce tariffs, and generally make Singapore an easy location for trade. This was crucial to expanding the scope of the economy, because with fewer than 4 million people (at the time), the domestic economy could not be sufficient to drive growth.
Attract foreign direct investment. As the economic infrastructure took root, Singapore became an attractive destination for foreign direct investment. The government assisted with courting and promoting foreign direct investment into the country, which provided investment capital to build out the economy. Tax and relocation subsidies helped a lot here.
Not all of these approaches will be applicable to Caribbean countries (e.g. Singapore's location was particularly strategic as a shipping hub), but not all of them need to be replicated in order to create fertile conditions for economic development in the Caribbean.