First, you must define the word first in your question. First ~ Chronologically. First ~ In importance. First ~ To be considered. Chronologically, writing would be one of the last arts for man to have invented.
Wouldn't it be something more basic, such as the first conscious act a child does of self-expression? It could be figuring out a way to get a toy from someone else by behaving a certain way. Just my rambling thoughts here.
the first 'writing' WAS painting [stories told in pictograph/cave art], so i guess those two arts are tied for 'first'... cooking isn't considered one of the arts, though it approaches an 'art form' at times... 'the arts' are considered to be music; art [includes painting, drawing, sculpture]; writing...
Qu: arts: studies intended to provide general knowledge and intellectual skills (rather than occupational or professional skills) If studying cookery does not intend to provide general knowledge and intellectual skills, I will eat my own feet. With a nice mushroom sauce and a bottle of Bourdeaux.