Picnic at a nice green shady park, not sure but maybe labrador, with bubbles and balloons to keep babies happy.
Alternatively, maybe at Sentosa building sandcastles.
Globetrotters in UE Square probably also do 1st birthdays - playroom and completely child orientated (food ok)
Get a friend to hire their function room on your behalf and have a bbq swimtime
It really depends on how many people your having and whether it is more a chance to catch up with your adult friends or your childs "friends". If it is the later, a small afternoon tea with balloons, bubbles and music would be fantastic fun for your little one and not too over the top for them.
When I looked into doing mine I did a google search on it and had some ideas through that along with some sensible advice.