Asymptotic form of Bessel $Y_0(x)$ for small $x$

The intuition is that for small $x$ only the contribution for "large" $t$ is important. Thus, we expect that $\sinh t \sim \tfrac12 e^{t}$. So we approximate $$Y_0(x) \sim - \frac2{\pi} \int_0^\infty\!dt\,e^{-x e^{t}/2}= - \frac{2}{\pi}\int_{x/2}^\infty \!dz \frac{e^{-z}}{z}. \tag{1}$$

Indeed, we have that \begin{align*}\left|Y_0(x) +\frac2{\pi} \int_0^\infty\!dt\,e^{-x e^{t}/2}\right| & \leq \frac{2}{\pi} \int_0^\infty \!dt\,\left| e^{-x e^t/2} - e^{-x \sinh t} \right| \\ & = \frac{2}{\pi} \int_0^\infty \!dt\, e^{-x e^t/2} \underbrace{(1- e^{-x e^{-t}/2})}_{\leq x e^{-t}/2}=O(x) \end{align*} such that $(1)$ is correct (up and with the constant term).

Starting with $(1)$, we can proceed as follows: $$Y_0(x) \sim I=- \frac{2}{\pi}\int_{x/2}^\infty \!dz \frac{e^{-z}}{z}$$ We calculate $$\frac{dI}{dx} = \frac{e^{-x/2}}{\pi (x/2)} = \frac{1}{\pi(x/2)} + O(1)$$ and thus $$I = \frac{2}{\pi} \ln(x/2) + C $$ with $C$ a constant.

The constant is given by \begin{align*} C= \lim_{x\to0^+}\left[I - \frac{2}{\pi} \ln(x/2) \right]&=\frac{2}{\pi} \lim_{x\to0^+}\int_{x/2}^1 \frac{1-e^{-z}}z - \frac{2}\pi \int_1^\infty \frac{e^{-z}}z \\ & = \frac{2}{\pi} \int_{0}^1 \frac{1-e^{-z}}z - \frac{2}\pi \int_1^\infty \frac{e^{-z}}z .\end{align*} I am not sure there is an easy way to show that $C = 2 \gamma/\pi$.

The constant might be fixed the following way:

Denote the integral in question is $D=\frac{\pi C}{2}=\color{blue}{J_1}+\color{red}{J_2}$ with

$$ \color{blue}{J_1}=\color{blue}{\int_0^1\frac{1-e^{-z}}{z}\quad},\quad\color{red}{J_2}=\color{red}{-\int_1^{\infty}\frac{e^{-z}}{z}\quad} $$

Let us start with $\color{blue}{J_1}$ and integrate by parts:

$$ \color{blue}{J_1}=\color{blue}{\lim_{\epsilon\rightarrow0}\left(\log{\epsilon}-\int_{\epsilon}^1\frac{e^{-z}}{z}\right)}=\color{blue}{\lim_{\epsilon\rightarrow0}\left(\log{\epsilon}-\log{\epsilon}-\int_{0}^1\log(z)e^{-z}\right)}=\color{blue}{-\int_{0}^1\log(z)e^{-z}} $$

Integrating now $\color{red}{J_2}$ by parts yields

$$ \color{red}{J_2}=\color{red}{-\int_{1}^{\infty}\log(z)e^{-z}} $$

therefore

$$ -D=-(\color{blue}{J_1}+\color{red}{J_2})=\color{blue}{\int_{0}^1\log(z)e^{-z}}+\color{red}{\int_{1}^{\infty}\log(z)e^{-z}}={\int_{0}^{\infty}\log(z)e^{-z}} $$

or

$$ D=\gamma $$

which implies

$$ C=\frac{2 \gamma}{\pi} $$