The fried noodles in the 1988 Korean drama are one of the traditional Korean cuisines, and I believe there are many restaurants in Changsha that serve them. For your reference, here are some of the more reputable restaurants in Changsha that I've learned about:

1. Little Sheep, No. 188, Bao Henan Road

2. Lao Changsha Fried Noodles, No. 58, Hejiazhuangzi, Daito Town, Station Street

3. Lao Guanping Fried Noodles, LG3-02, Juntun Vanguardi Shopping Center, Tongzipo South Rd. Southwest

Of course, the above are just some of the more famous restaurants, you can try searching for more. Meanwhile, due to the change of time and market situation, whether the above restaurants still serve fried noodles now or not still need to check the restaurants specifically.