I have a 4 foot by 6 foot area of Zoysia grass that I want to remove from my regular lawn.

1. Roundup or dig it up or both?
2. In Nebraska, Is it best to remove this grass in Spring or Fall?

The sod will have to be actively growing for Round-Up or other similar systemic herbicides to work effectively. I would wait until it is no longer dormant, Round-Up it and then dig it out a few days later and install new sod of your choice. You can't just lay new sod on top of old if it is sod you are installing. The new sod will sit too high and won't root in properly. It should be in contac with the soil. If the new sod has 1" of soil on the bottom you need to regrade 1" lower than the existing sod so the new will fit smoothly. Remember to not apply pre emergents to new sod. I didn't know Zoysia grew as far North as Nebraska. Learn something everyday.