First off...be VERY VERY careful with the starting fluid. It can wash the tops of pistons if it doesn't have upper cylinder lube...and can hurt a two storke in a hurry.
Lets look at the basics....you have to have air/fuel/fire....to have an engine run...take away one of these...and you have nothing.
You obviously have fire...and air...because it starts and runs.
This leaves us with fuel. You said you replaced fuel pump. Well I'm guessing you had problem before the pump...this is why you replaced the pump. That being said...sounds like you are loosing your prime.
This can happen for a few reasons...most likely...that I've seen...you need a new bulb and fuel line. Over time these will get to where they do not pump...mainly due to valve failure inside the bulb...when you pump...it doesn't hold prime...it just forces fuel back and forth...kind of like an antidrainback valve.
If this is okay...check the fuel filter...there should be a screen type filter under the cowling somewhere. They are normally little white plastic looking canisters...trace the fuel line and you'll find it.
Probably the easiest and quickest thing to check is your vent. If your vent is clogged or closed or whatever...you will have a kind of vapor lock situation. Without the ability of air to come into the tank and replace the fuel that has come out you will loose prime and fuel will lock in the lines and not move due to the vacuum.
If all this is okay...pull that carb and rebuild her. You can get a kit LOTS of places online and rebuild it yourself. Those have one carb are normally easy to get to and there isn't much to them. Probably be a good idea to do it anyways if the condition of it is unknown. Have you run any bad or old gas through it?? Did it have ethanol in it? Check out Canepoles threads about ethanol.
Since you are able to start on starting fluid I'm guess she stays running after you get her started. This being said...I'm really thinking you have a priming problem. Once the engine is running you pull fuel...but cranking alone isn't pulling hard enough to get it to the carbs...but 1000rpm is...cranking is normally 300 or a bit better. Just some thoughts to check out. If these things do not get you anywhere...let us know.
"But if serving the Lord seems undesirable to you, then choose for yourselves this day whom you will serve, whether the gods your ancestors served beyond the Euphrates, or the gods of the Amorites, in whose land you are living. But as for me and my household, we will serve the Lord.” Joshua 25:15