We just leave the repeated phrase out: “I’m getting my car repaired.” Does that not work in Korean? If you really feel like you need something, then you could say “then”. By the way, we use the future with “going to” to talk about plans. We use “will” to talk about decisions in the moment, so if you say “will”, it’s like you decided to get your car repaired then *after* they asked if you were free. BTW: "The day after tomorrow" is a specific day, so it takes the definite article.

Use 'then' to avoid repeating the same date. When you use it, it's assumed you are referring to the date just mentioned. "No, I'm not. I'm getting my car repaired then."