Now that it's sheltered, it will start growing as if its season has just started. I don't remember how long a chrysanthemum takes, but it should be ready by the next day. Grafting has to be done to a mature plant, so that's why you can't do it yet, you just have to give it time to grow. Your cutting should be fine until you use it, they never go bad.
Your sim can take comfort in the fact that his investment will pay for itself. With his garden always in season, he'll have a lot more produce to sell!
Be aware, once you do the graft: The chrysanthemum plant will immediately revert to a just sprouted stage. That's normal. Then it will grow normally again, and now it will start producing chrysanthemums and tulips every day. And a random chance for a bird of paradise. Rarer plants produced by grafting can take several days before you get one, depending on how lucky you are.
You can also, before grafting your tulip cutting onto the chrysanthemum, take a cutting of the chrysanthemum first. You can graft that cutting to your tulip, and have TWO plants with a chance of producing a bird of paradise for you. It doesn't matter if it's a cutting from tulip grafted to chrysanthemum, or vice versa, the combination of tulip and chrysanthemum will get you a bird of paradise. It works the same way with any other grafting too, it's the combination that gets the new plant, doesn't matter which is the donor and which is the recipient.
Oh, and if you have City Living-- be sure to attend the Romance and Spice Festivals when they come up. There will be Perfect quality plants to pick! The Romance Festival will have Lilies when they're in season, I always have a hard time getting my hands on one of those. And they're one of the things needed when you're trying to get Death Flowers. I've heard a lot about what a pain it is to get a Pomegranate too, but I just buy one off the gifts menu. The Spice Festival will have vegetables to pick, and even one grafted plant which will give you a Cowplant berry! Be careful with those. When they get hungry, they can kill someone. Also, when they get hungry, they need to be fed within about 12 hours or they die. Dead Cowplants do make a pretty cool decorative object though.
Another thing, if you have City Living-- find the Produce vendor. There are spots all over the city where vendor stalls pop up, and it's random which ones will come up. You might have to visit the city on a few different days before you find it. Most of them sell food, but the Produce stall will sell you fruits and vegetables. Maybe flowers too? I don't remember. It's a lot cheaper to buy one of something you don't have already, than to buy lots of seed packets hoping to get something.
