Dear Bets, congratulations on venturing south and experiencing some of the world’s oldest cultures in the Asian sub-continent.
Singapore is a modern and vibrant city, exceedingly clean, but can be overly humid and hot for many. My wife struggles with both, I find the humidity can leave one feeling rather fatigued at the end of the day. Best fixed with a few beers. Keep yourself well hydrated with water. Actually, this will apply for the duration of your trip. The mass transport system (subway) is underground with large mall type buildings on top. Taxis are cheap, reliable and honest. Accommodation can be expensive while food prices vary greatly. Lonely Planet books/web site, or similar should have good recommendations and up to date information about food and the local tourist sites to suit your taste. Your cruise line may also provide/suggested resources. Great views from Marina Bay Sands at night, as always comes with a price. With afternoon tea at the Raffles one is closeted in a colonial era dining room within a magnificent building and surrounded by history, also for a price. (My wife loves the place.)
Assuming you are in Dubai around April, the climate should be hot, with humidity increasing as summer approaches. I like the heat. We often stop at Dubai for a day or two when returning home from Europe. Normally in October. As a halfway point it helps to mitigate any jetlag. We take an apartment closed to the beach. Great swimming, water is warm and flat. In the UAE lots of things for young adults and teenagers with shopping skewed towards luxury brands. Magnificent airport.
In Dubai, western women are not required to wear scarves or other head covering. Unless you are planning to meet with the Emir. As a guide, just think you are going to your local Episcopalian Church during summertime. Consult lonely planet guide or similar for accurate clothing advice.
Yes, the toilet facilities can be a delicate subject. I venture the onshore tour operators will have it under control. Acknowledging that I am a bloke, I note that during the East India Company times (pre white squatters) women accompanied their husbands to this region. Don’t know if any wrote of their experiences. Probably not the type of thing Georgian/Victorian era women wrote about. They do appear to have survived the experiences.
Enjoy the cruise and new cultural experiences and I hope you get to see a full moon rising while at sea.