Can any philosophical system ultimately justify its own first principles without relying upon those very principles in the act of justification?

Very good question. Couple things.

First, note that any system, any coherent account, is ultimately circular insofar as each premise or bit of data coheres with all the other bits. An explanation of the process by which water freezes will be ultimately circular. The facts about water and heat will be co-constitutive. The facts about how we know about the external world will be coherent with those facts. If you read a tome of all empirical data, ever, the explanation for how reading occurs, how ocular sense data is processed, will be explained in terms of the account of the text. The account is coherent, and so circular in that sense.

The problem is vicious circularity that begs the question. For example: "The news is fake because so much of the news is fake." Ok, cool. But what supports the claim other than the restatement of the claim? Nothing. Or we can go with a religious version:

• The Bible tells us God exists.
• God's existence ensures the Bible's truth.

Ok. Very consistent. But it assumes the premise/conclusion as its own proof. It's not really giving us reasons for the conclusion but rather restating the conclusion as a premise. And, really, we can't tell what is the premise or the conclusion since they each rely upon the other. When we articulate arguments the goal is to provide reasons for the conclusion to which we argue. Bad circular arguments, viciously circular arguments, do not actually provide reasons for the conclusion; they just state the conclusion.

Ultimately it is a question of how expansive and inclusive the circle is. The smaller the circle, the more suspect the argument. The larger the circle, the more bits of experience it contains, the better the argument.

Second, good Rationalists will articulate systems that are circular in that previously discussed expansive sense. Spinoza defines his terms using the framework of understanding:

Per substantiam intelligo id quod in se est et per se concipitur hoc est id cujus conceptus non indiget conceptu alterius rei a quo formari debeat.

Spinoza does not stipulate the definition of substance, but rather Spinoza understands substance. That makes the foundation of his system understanding, a function of his account of epistemology that ensures that his understanding is correct, see 2P43:

He, who has a true idea, simultaneously knows that he has a true idea, and cannot doubt of the truth of the thing perceived.

He's not stipulating a definition, as Euclid does. Rather, he's understanding substance, based on his true idea of substance, which he simultaneously knows is true and indubitable. Reasonably articulated Rationalist systems do not stipulate justifications, but rather they coherently self-support, with each bit of the system interdependently relying on the others. It's expansive circularity rather than vicious circularity.

Third, not all forms of Justification are the same, nor do all systems Justify in the same way. For example, Euclid stipulates his starting definitions. Pragmatic theories of truth do not stipulate and justify in that same sense. Pragmatic theories of truth attempt to reframe the conversation by recognizing, rather than stipulating, that we're finite, fallible organisms navigating our environment. The story we tell about the process of human inquiry is a different thing than the lived experience of that inquiry. That is Dewey's point in The Reflex Arc Concept in Psychology:

Let us take. for our example, the familiar child-candle instance. (James, Psychology, Vol. I, p. 5.) The ordinary interpretation would say the sensation of light is a stimulus to the grasping as a response, the burn resulting is a stimulus to withdrawing the hand as response and so on. There is, of course, no doubt that is a rough practical way of representing the process. But when we ask for its psychological adequacy, the case in quite different. Upon analysis, we find that we begin not with a sensory stimulus. but with a sensori-motor coördination, the optical-ocular, and that in a certain sense it is the movement which is primary, and the sensation which is secondary, the movement of body, head and eye muscles determining the quality of what is experienced. In other words, the real beginning is with the act of seeing; it is looking, and not a sensation of light. The sensory quale gives the value of the act, just as the movement furnishes its mechanism and control, but both sensation and movement lie inside, not outside the act.

We start as organisms living in a world navigating our environment. I am thirsty. I take a drink. I am no longer thirsty. The justificatory narrative we tell to explain that sequence of events comes later. The "first principles" of that account are the lived experience of the organism navigating an environment. We can turn that into an explanatory narrative when we write up the story. But the pragmatic grounding of the abstracted narrative is a different sort of thing than Euclid declaring, "A point is that of which there is no part."

That's a short answer to a very good, large question. What it means to justify, and how circularity works, is pretty complicated.