The Fullerton will be rather short on outdoor space, as it is the former colonial post office right in the middle of a high-traffic area of the CBD. For "old school classy" with appreciable outdoor space, you might consider Raffles. Definitely a splurge hotel. The entire interior of the Raffles complex is beautifully gardened courtyard.

I'd focus my "outdoor" fix on Bali, and in Singapore accept you are primarily in an urban setting. I personally like the Singapore Marriott, but I've stayed there numerous times and have high status there. Excellent service though, and right on Orchard Road. I think the Hilton is a bit tired and needing renovation.

Don't worry too much about location - anything and everything in SG is easily accessible by taxi.

For reference and comparison, what are your Bali plans? I would enthusiastically recommend Hanging Gardens Ubud.