The reality of your situation is this: You probably should rebuild these carbs.
Even if you can overcome the starting problem (getting the fuel flowing again after years of sitting in storage) I'm betting there are other issues just waiting for their turn on stage.
There's a reason why this Bandit was parked a few years ago. Probably because over time it had become harder and harder to run. Eventually, when it became really difficult to deal with it got pushed to the back of a garage and forgotten.
The Mikuni carbs on the B4 are good but they require attention. Your carbs are 24 years old now. You should replace all the old rubber parts. Here's the guy who can provide you with all the rubber bits:
http://www.litetek.co/Carb_Kit_Suzuki_GSF400.htmlWhile replacing all the rubber bits with high quality Viton is a great move, there might be another issue.
Some of the brass parts of carbs wear with age. In the B4 carbs the most common worn brass part is the emulsion tubes. The emulsion tubes can become eroded into an oblong shape on their inside rather than perfectly round because the needles drag against them as they move up and down.
The larger gap around the needle caused by eroded emulsion tubes causes too much fuel to flow into the air/fuel mixture, no matter how you try to tune them otherwise. This condition causes a B4 to turn into a hard-to-start, poor-idling, over-rich pig. Which, as mentioned earlier, often causes it to be abandoned in the back of a garage.