How are first digits of $\pi$ found?

You make the assumption there is no way to calculate the digits of $\pi$. That is untrue; there exist many formulas to calculate the digits of $\pi$ (with their own proofs of correctness). One of the simplest (though very slow) formulas is the Leibniz formula for $\pi$.

One way the ancient mathematicians used was to compute the perimeter of a regular $n$-gon inscribed in a circle of diameter $1$. If one finds a formula for the perimeter of the inscribed $n$-gon, one has a sequence converging to $\pi$, i.e. for any accuracy you require there exist an $n$ such that this formula is accurate enough. This is not very practical though. In reality these ancient mathematicians such as Zu Chongzhi used iterative methods to compute the first digits of $\pi$.

From a more modern perspective, in analysis one seldom defines $\pi$ as the ratio of the circumference of the circle to its diameter. Here is one way: Let $\exp:\mathbb{C}\to \mathbb{C}$ be a function that satisfies $\exp(x+y)=\exp(x)\exp(y)$ and the derivative at $0$ being $1$. Then one can prove that such a function is unique and is realized by the function $\exp(z)=\displaystyle\sum_{n=0}^\infty \frac{z^n}{n!}$. One then defines $\sin(z)=\displaystyle\frac{\exp(iz)-\exp(-iz)}{2i}$. Consider its restriction on $\mathbb{R}$, then $\pi$ is defined as the smallest positive real number $x$ such that $sin(x)=0$. Then we immediately have very strong tools to compute $\pi$.

That a number is irrational means only that it cannot be computed merely by dividing one integer by another. Square roots of integers that are not perfect squares are irrational, but they can be computed by other methods that are just a little more complicated than long division. (These methods come down to addition, subtraction, multiplication, division, and sometimes a limited number of trial-and-error steps.)

Methods of computing the digits of irrational numbers simply find some computable quantity that is known to be very close to the true value of the desired number. If you have a formula that is known to be within the bounds $x \pm 0.0005$, for example, you merely need compute the value of that formula to enough places that you can accurately round it to the nearest 0.001, and then you know for sure the first three digits of $x$ to the right of the decimal point. To compute the first $100,000$ digits of $\pi$ requires you to evaluate a formula known to be within the bounds $\pi \pm (0.5\times10^{-100000}).$ This is difficult but not impossible.