Before putting the carbs back on the engine, get a gas/oil mix into each of the float bowls as full as you can get them. Also, before connecting the cleaned out fuel lines to the carbs, get some gas/oil into each of them and hopefully the fuel pump will be able to push this into the carbs when the engine starts.
An easy way to do all of this is with a Zoom-Spout oil bottle (google if you're not familiar with these). Get an empty Z-S bottle, or dump a full one into another container so you can fill the bottle with gas/2 stroke oil. The little spout on the bottle fits right into the fuel lines (I'd run the spout right up to the fuel pump and squeeze the gas mix in, then suck it out and squeeze in again, etc, as this may help thin out the oil in the pump if that's what it is full of.
Also to fill the carb bowls, remove the little drain screw in the bottom of the bowl, tip the carbs a bit and plug the drain nipple with your finger (or whatever you have available) to keep fuel from running out, and stick the spout into the hole where the screw was. Squeeze the bottle slowly to get some gas mix in the bowl, and replace screw in bowl. Don't tip the carbs too far during or after filling, as fuel may find it's way out of the carb or bowl vents. Repeat for other carb, and when done keep carbs relatively upright until installed on engine. This works very well for me every time I clean carbs, and the sled usually starts in a few pulls since the fuel pump doesn't have to fill empty float bowls from me tugging on the recoil rope to "energize" the pulse line.
In the absence of a Z-S bottle, substitute whatever you have available to try to accomplish the same as the above. Good luck.