The 737-900 has the shortest range capabilities of all the 737NG's...
If you want to fly far away with a 737...choose the 737-700.
With 110pax, it can fly from Boston to Berlin, although you'll be flying on fumes when you arrive
To answer your first question, YES...You can fly from LA to Honolulu with the 739, but you'd better pick "as the crow flies" direct routing
Straight line between LAX and PHNL measures 2217 nautical miles
Max. Range for the 737-900 is somewhere around 2.700nm, so pray that you don't have a strong headwind (more fuelburn) and travel light so you can cruise on high altitude. Long Range Cruise (LRC) speed for the 737's is around mach .76
You wont' make it between Honolulu and Tokyo though (not even with the 737-700) but flying between NY and Barbados should be just fine
Although, like Gary pointed out, it's not common to see the 737 do these long legs, if they do it's only Business Jets I assume. The aircraft has to be rather light with passengers and heavy on fuel load, and that's not economical at all.