What Arduino do you want to use? Each LED requires 3 bytes of SRAM so that is 3000 bytes of memory, so you need an Arduino like the Mega or one of the Arm powered Arduinos for that.
Going from 343 LEDs to 1000 is a big jump and you will have to pay more attention to your power supply decoupling than you did with your smaller project. You are also talking about a much bigger power supply, you need on that will provide 240 Watts @ 5V so that would be 48A. Where as you "only" needed 16.5A before.
For my 343 LEDs I used an Arduino Nano but for some animations 2KB were not enough wherefor I did use an Arduino Mega but even 8KB is a bit small for complicated animations. A Mega would not be enough for 10*10*10. I tried using an STM32 but their implementation is a bit glitchy and did not work for me. I think I will use an Arduino Zero. With 32KB I wont run out of RAM.
I measured a maximum current draw of 42,8mA per LED at RGB 255|255|255. I think the current draw would be arround 42,8 * 1000 = 42,8A. Therefor would I use three 20A power supplies cooled by ultrasilent fans that I have laying arround from my PC. The Voltagedrop was with my 7*7*7 Cube at max -0.3 or min. 4.8V. Because I would have three power sections will this be no problem.
For Decoupling will I use 1000 100nF SMD Ceramic Multilayer Capaitors soldered a few millimeters away of the legs of every LED.
I am just wondering if it is possible to pass a data signal consisting of 1000 24Bit through the LEDs.