Sitting 26 years and now running like the carbs are gummed up... hmmm.
Probably the carbs are gummed up.
You may also have ignition arcover troubles, the plug cables deteriorate with age and tend to spark through the insulation to metal things close by. Once it runs, starting it in a very dark garage and looking up under the front of the engine where the coils are lets you see the errant sparks if there are any. Don't do this carelessly and die from exhaust poisoning, by the way.
The carbs tend to get dirty even when drained and stored inside. Humidity alone causes a bit of corrosion over a couple of decades. You have to remove the carb rack to do this, but shouldn't need to disturb the sync adjustments unless the emulsion tubes are really stuck. You have to pull the slides out if heavy persuasion is needed to get the emulsion tubes out but they will need to be cleaned and checked: trying to apply the force through a throat will pretty much always cause damage to the tube, body, slide, or needle. It can be done with a bit of wood dowel using the slide as a "hammer" to tap the tube down but you have to be very careful.
Otherwise it's just remove the jets, airscrews, and float valve and clean everything including the tiny cross holes in the pilot jet and the main jet emulsion tube. Blow carb cleaner and compressed air through all the body passages until they seem open and all four carbs blow out about the same from the same places. Avoid reaming out any holes with wire or pipe cleaners (the twisted wire core will damage the soft aluminum and brass). Honda sells gasket kits, these are (in my opinion) better quality than the aftermarket "rebuild kit" kit gaskets. Avoid using aftermarket kit jets and needles as a rule. Float valves are available from Honda but the aftermarket ones are pretty good.
If fuel was left in the carbs 26 years ago you'll find a crust of gunk in the float bowls. If it was parked with fuel in the tank and the petcock on, you will find a horror show in the float bowls.
