Basic troubleshooting -
Healthy Spark? If not, install new plug, check points. If points are funky, clean up, then check timing.
Carburetion - DO NOT depend on the tap filter. There can be no dust, chunks, crud or water in the float bowl. Use an aftermarket clear filter in 10 - 40 micron range. If it doesn't specify the microns, get one that does.
Carb mount - use 1 1/4" fiber reinforced radiator hose & 2 standard hose clamps. MUCH more sturdy & durable. Cracking in the carb to head connection makes diagnosing trouble harder.
IF there was crud in the float bowl, disassemble & clean. Take out the jets & back blow all passages, WD40 is handy & clean. You should be able to see thru the pilot jet, main jet, needle jet and their emulsion holes. CLEAN is good. Check float level; nominally 24mm - 26mm ( 1"). Make sure the float isn't perforated or saturated & full of gas. Don't forget the Air screw; blow out and set back to where you found it. normally in the 1 1/2 - 3 turns from lightly bottomed range.
With good spark, a clean carb and clean fuel to the carb, it should start & run.
Occasionally an old coil or condenser will quit working when it warms up. If it quits whilst running, immediately check the spark. If unsure, replace the coil & condenser and be done with it. The carb is pretty idiot proof if you keep it's internals clean. Running a screen vs. an air cleaner was SOP for years, but dust/road crud can foul plugs and will accelerate engine wear. Probably not a big deal on the Bullet, just carry extra spark plugs.