The famous theorem of Hadamard and Vallee-Poussin https://en.wikipedia.org/wiki/Prime_number_theorem implies that $p_n\sim n\ln n$, so $C_1 n\ln n \le p_n \le C_2 n\ln n$ holds for all $n\ge 2$ with some constants $C_1,C_2$.
Since $p_n\sim n\ln n$, we obviously have $C_1\le 1$.
I found somewhere in the internet (I don't remember the exact source) that $C_1=1$ and $C_2=29$ works fine.
How these values were derived?
Is $C_2=29$ optimal or a smaller value is known?
The lower bound, that $n \log n < p_n$ was proved by Rosser in 1938, and is sometimes called Rosser's Theorem.
For upper bounds, the indisputable expert is Pierre Dusart, who has several results revolving around explicit bounds related to the sizes of primes and collections of primes. He proved that $$ p_n < n \log n + n \log \log n$$ for $n \geq 6$. In particular, $n \log \log n < n \log n$, so the bound $C_2 = 2$ should be fine for $n \geq 6$. If you are interested in particular for the bound holding for all $n \geq 2$, then it's just a matter of checking for $n \leq 5$. A quick numeric check indicates that $C_2 \approx 2.16405$ suffices for $n \geq 2$.
