You'll probably have no difficulty learning Karate and Judo at the same time.
The danger in studying more than one martial art at the same time is that you can confuse the two. When the martial arts are more alike, it's easier to confuse them. When they're more dissimilar, they're easier to learn together.
For example, practicing Shotokan karate and Taekwondo at the same time could be confusing. Both have roughly the same stances and techniques, because TKD comes from Shotokan. But TKD changes everything in subtle ways. The stances in TKD are more upright than in Shotokan. You will also notice that the center of gravity in a TKD practitioner bobs up and down, unlike in Shotokan. The sparring rules are different between the two. And the emphasis Shotokan has on one highly focused, fast, clean attack is tossed out the window in TKD. As a result of all these differences and more, it's easy for you to confuse the two and find yourself doing a technique in a TKD way while in Shotokan class, or vice-versa. Your instructors will realize you're training in another martial art at the same time, and they will ask you to stop doing the other one in order to learn the first one.
It's confusing learning Judo and BJJ at the same time, too. BJJ sorta comes from Judo. They share a lot of techniques. But like Shotokan and TKD, the two emphasize different things and have subtle differences that make it hard to learn them both at the same time without confusing them.
But when you have two very dissimilar martial arts, it's easy to learn them both without confusing them. In your case, Shotokan karate and Judo are completely different. You won't have any issues confusing them. They are perfectly suited to be learned together. And yes, they complement each other well.
You'll also learn later on that Shotokan karate kata is composed of grappling primarily. All those blocks and strikes aren't necessarily doing what you think they're doing. You'll find that there are judo-like throws in the kata.
I go over this in another answer, which you can read at the following link:
Name and meaning of stance where you stand with fists on hips?
Another approach you might consider is to learn classical Japanese jujitsu instead of Judo. The knowledge you gain in jujitsu would help you understand your karate kata to a much greater degree than Judo would.
What Judo practice gives you, though, is the ability to fight. It's how you train that matters. Training against someone who is determined to beat you and who will resist everything you do is the way you gain fighting ability. BJJ would also give you that. But both Judo and BJJ would not give you as much insight into your karate kata as classical jujitsu would. So it just depends on your goal which one you decide to learn.
Next, you have to consider your time. Most people who do two martial arts eventually burn themselves out. They find that they just don't have the time to do both well. So they drop one and go with the other. And actually, eventually they give up martial arts altogether when their lives get really tight for time.
It sounds like time is not your problem right now, though. In that case, the risk for you would be overtraining. Overtraining means you workout too many times a week and for too long. Your body may not get a chance to recover. And so your progress over time is hindered. Keep that in mind. You might do better scheduling both martial arts practices on the same 2 or 3 days a week. Then put one or two days of rest between those days. That way, you can recover in time for the next class.
Hope that helps.