As a junior developer I've been on two projects at once, One was a rather big project that had the whole team working on it but the second one is a smaller project that only I am working on.
I'm finding the smaller project incredibly difficult as I'm talking to the client, coding and having to manage my time by myself. I feel that the responsibility is too much.
So my question is, How can I tell my managers that this amount of responsibility is too much for me without it making me look bad?
You can't cry vaguely about the whole project, it will make you look bad.
What you can do is focus on your strong points whatever they might be, and attempt to get your manager to give you assistance with the others.
Unfortunately for you, the three things you mentioned, time management, customer relations and coding are all core skills that even a junior should be able to handle unless expectations are unrealistic, in which case focus on what needs improvement, rather than blame yourself.
So find what in particular is causing you grief and firstly attempt to resolve it yourself, and if it's outside your control, speak to your manager about it. Quite often just identifying and working on issues methodically like this is a solution in itself.
Make the effort, this is the sort of project that can springboard your reputation and career (or severely retard it). Do not think of it as a small project, it's not, it's the most important project you have had. It's also very important to the client, they're outlaying money on it. To them it might be the most important IT project of the year and you got your name all over it.
It's easy to feel overwhelmed when you're new, cut the issues down to size and soldier through them is what worked for me. Don't be shy to ask the manager for help, but with specifics not generalisations. Document everything you do for reference and be thorough... don't let things go South, if you get stuck and really need an assist, don't sit on it, ask for help, solve that problem, move forwards to the next.
To echo what Kilisi has said above, as a junior, you shouldn't be expected to manage your time to the same level as a senior developer. If you feel as though you're being over-worked or you're finding it difficult to complete your tasks, then speak with your management team about it (they're there to manage you as a resource, after all).
You can work this into a positive move for yourself if you ask for more time for you to be allocated to this side project and through that demonstrate that you're building skills in managing/implementing that project.
In any case, talking to your manager is key, and agreeing some time split between these projects will give you some structure.
You won't look bad for talking to your managers. You might well look bad if you don't (and things subsequently spiral out of control...)
It is good to talk to your managers and get ahead of it by taking responsibility. As a Junior Dev, you shouldn't be too responsible for setting your own priorities as much as a Senior Dev, being that you are still relatively new and needing to get a feel for the operational tempo of your particular organization.
Most of what you are doing are core skills - maybe not so much client relations, but that totally depends on your companies structure. I am a program manager, and as a project manager before that I wouldn't let the clients talk to the Devs - that is not supposed to be your job, but alas, if you don't have a PM or Business Analyst - or some other buffer - that is going to be your responsibility unfortunately.
You have to mentally prepare yourself to not let the feelings of being overwhelmed impact you, but laying all of this out for your manager without coming across as petulant and whiny should go just fine, I would be very inclined to come to a compromise if I was your manager. It wouldn't hurt to cite quality of work will go up if your level of effort wasn't split across multiple domains of multiple areas.
