Very safe in Singapore and reasonably priced too. I don't know about Shanghai as I have never been there, hope to one day. Have a lovely cruise and give my love to Singapore, I had a wonderful life out there and miss it like mad.
Certainly taxis are easy to find and relatively inexpensive. Not sure of your itinerary but I am going to assume that until you reach Shanghai English is spoken fairly frequently. I would suggest that you look at the excursions offered by the ship and then pick out what you would like to see. You then research these tours on the www looking at the area's own tourism pages. There are other sites where you can find those sailing on the same cruise and make contact with them. So work out your own tours and then hire taxis. Shanghai is different for although some English is spoken it is not too prevalent and unless things have changed recently it is difficult to DIY. I do remember dancing in the streets round the Bund but can't remember the day of the week but the Chinese love doing this and it was great fun. Two phrases that may come in handy........Nee How Ma ....How are you ..... and ... Een Pee Joe . One cold Beer I would recommend seeing the old town, by excursion in Shanghai that was a good trip. Good Luck....... Neil.