Depending on the environment, it is possible to go over winter on the veranda etc.
As I want to create an environment suitable for plant cultivation as much as possible, I did not grow curry leaves outside the room at low temperatures until 2019-2020 winter. That was the very first winter for me to overwinter a pot of curry leaf in my veranda.
Over here in Japan where I live, western suburb of Tokyo, it looks like you should be able to overwinter the curry leaf in most of the Hardiness Zone 9b and 9a.
You need to harvest the leaves first! As the temperature drops, the leaves will lose their green color, also its fragrance gets lost as well, so it is important to harvest them first. Snap the petiole (the part that looks like a twig) from its base to the trunk of the plant.
Prune early to get the next year’s buds to shoot sooner. If you prune later into mid spring or early summer, you will have the new buds come out from a place where they haven’t yet differentiated, so the plant will need some time to differentiate to make new buds so that you will have to wait to enjoy leaves until later.
Placement and management of pots: Always avoid areas where the soil can be frozen, but it is better if they are in a sunny place or near a wall so that they are exposed to radiant heat from the wall.
Fresh leaves can be stored in fridge for a few days in air tight plastic bag. If you intend to use the leaves for cooking, frying the leaves will let you enjoy the aroma of the leaves for a couple of months by storing the fried leaves in the air tight container. Here is a link to the process
Video of frying the leaves to preserve
During the winter months
Watering is necessary to prevent the soil from becoming overly dry, but since the plant has no leaves, its water absorption and evaporation from the plant is limited.
You can use BBQ bamboo stick to check if the soil is moist or dry. If it is moist, you will see the color of the bamboo stick turned to darker. Then you do not have to water the plant.
If you haven’t pruned them yet, prune them so they don’t get a late start when spring arrives, and you might want to avoid to prune them in the middle of winter, which is why you need to prune them sooner rather than later.
When it gets warm
If you haven’t pruned your Curryleaf, you will experience apical bud dominance (tip dominance), which may increase your psychological resistance to pruning and the waste of sprouting new shoots.
Be careful not to damage the sprouts. The sprouts will flop off from the base of the plant if you touch them just a little bit. The sprouts can come off even if part of clothes touches them. (I’ve experienced more than a few times).
We’ll wait patiently for the sprouts, and when the highs are around 20 degrees C, we hear about new buds shooting, but even then, you may not see any sprouts, and my Curry Leaf #2, which overwintered on our balcony in winter of 2019-2020, didn’t have new buds until early June! It’s about an 8-year-old plant. But at the end of May, I found a greenish protrusion and a few days later I could see it as new buds.