$\require{AMScd}$I'll expand a little on my comment to give an answer to David's follow up question:

Firstly, the general relationship is described in Chiarellotto's Duke 1999 paper *"Rigid cohomology and invariant cycles for a semistable log scheme"*. He shows that the weight-monodromy conjecture implies that there is an exact sequence
\begin{equation*}
H_{rig}^{n}(X_{k}/K_{0})\rightarrow H_{log-cris}^{n}(X_{k}/K_{0})\xrightarrow{N}H_{log-cris}^{n}(X_{k}/K_{0})
\end{equation*}
This is unconditional if, for example, $X$ is a semistable family of curves or surfaces. In these cases the first map is an injection for $n=1$. For searching purposes, the elements of $\ker N$ are often called *"$p$-adic invariant cycles"*.

As for your second question, Chiarellotto states in the same paper that one cannot expect the first arrow to be an injection for $n\geq 2$ in general. Here is my guess as to why:

For simplicity, let us suppose that there exists a proper scheme $P$ over $W(k)$ such that $X\hookrightarrow P$ and $P$ is smooth around $X_{k}$. Let $X_{k}=\bigcup_{i\in I}X_{k,i}$ be the decomposition of $X_{k}$ into irreducible components. Then $\left\{]X_{k,i}[_{X_{K_{0}}^{rig}}\right\}_{i\in I}$ is an admissible covering of the rigid analytic space $X_{K_{0}}^{rig}$ associated to $X_{K_{0}}$, and similarly $\left\{]X_{k,i}[_{P_{K_{0}}^{rig}}\right\}_{i\in I}$ is an admissible cover of $]X_{k}[_{P_{K_{0}}^{rig}}$. The Čech cohomology of the first cover computes $H_{log-cris}^{\ast}(X_{k}/K_{0})$ and the Čech cohomology of the second cover computes $H_{rig}^{\ast}(X_{k}/K_{0})$.

Let us call the first map in above sequence $\psi^{\ast}$; it is the map on cohomology induced by $\psi:X_{K_{0}}^{rig}\rightarrow ]X_{k}[_{P_{K_{0}}^{rig}}$. It induces the following commutative diagram where the rows are the Mayer-Vietoris sequences for the two coverings:

\begin{CD}
\displaystyle\bigoplus_{i\in I}H_{dR}^{1}(]X_{k,i}[_{X_{K_{0}}^{rig}}) @>\alpha>> \displaystyle\bigoplus_{i<j\in I}H_{dR}^{1}(]X_{k,ij}[_{X_{K_{0}}^{rig}}) @>>> H_{log-cris}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{dR}^{2}(]X_{k,i}[_{X_{K_{0}}^{rig}}) @>\beta>> \displaystyle\bigoplus_{i<j\in I}H_{dR}^{2}(]X_{k,ij}[_{X_{K_{0}}^{rig}}) \\
@A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA\\
\displaystyle\bigoplus_{i\in I}H_{dR}^{1}(]X_{k,i}[_{P_{K_{0}}^{rig}}) @>\alpha'>> \displaystyle\bigoplus_{i<j\in I}H_{dR}^{1}(]X_{k,ij}[_{P_{K_{0}}^{rig}}) @>>> H_{rig}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{dR}^{1}(]X_{k,i}[_{P_{K_{0}}^{rig}}) @>\beta'>> \displaystyle\bigoplus_{i<j\in I}H_{dR}^{1}(]X_{k,ij}[_{P_{K_{0}}^{rig}}) \\ \end{CD}

Since the $X_{k,i}$ and $X_{k,ij}:=X_{k,i}\cap X_{k,j}$ are smooth and proper (by the semi-stability assumption), we have $H_{dR}^{\ast}(]X_{k,i}[_{P_{K_{0}}^{rig}})\simeq H_{cris}^{\ast}(X_{k,i}/K_{0})$ and $H_{dR}^{\ast}(]X_{k,ij}[_{P_{K_{0}}^{rig}})\simeq H_{cris}^{\ast}(X_{k,ij}/K_{0})$. If we give each $X_{k,i}$ the log-structure induced by the divisor $D_{i}$ given by intersecting $X_{k,i}$ with the other irreducible components, and similarly for the $X_{k,ij}$, we have $H_{dR}^{\ast}(]X_{k,i}[_{X_{K_{0}}^{rig}})\simeq H_{log-cris}^{\ast}(X_{k,i}/K_{0})$ and $H_{dR}^{\ast}(]X_{k,ij}[_{X_{K_{0}}^{rig}})\simeq H_{log-cris}^{\ast}(X_{k,ij}/K_{0})$. If we let $U_{i}$ be the complement of $D_{i}$ in $X_{k,i}$, and similarly for $U_{ij}$, we have $H_{log-cris}^{\ast}(X_{k,i}/K_{0})\simeq H_{rig}^{\ast}(U_{i}/K_{0})$ and $H_{log-cris}^{\ast}(X_{k,ij}/K_{0})\simeq H_{rig}^{\ast}(U_{ij}/K_{0})$. The above diagram is then

\begin{CD}
\displaystyle\bigoplus_{i\in I}H_{rig}^{1}(U_{i}/K_{0}) @>\alpha>> \displaystyle\bigoplus_{i<j\in I}H_{rig}^{1}(U_{ij}/K_{0}) @>>> H_{log-cris}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{rig}^{2}(U_{i}/K_{0}) @>\beta>> \displaystyle\bigoplus_{i<j\in I}H_{rig}^{2}(U_{ij}/K_{0}) \\
@A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA\\
\displaystyle\bigoplus_{i\in I}H_{cris}^{1}(X_{k,i}/K_{0}) @>\alpha'>> \displaystyle\bigoplus_{i<j\in I}H_{cris}^{1}(X_{k,ij}/K_{0}) @>>> H_{rig}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{cris}^{2}(X_{k,i}/K_{0}) @>\beta'>> \displaystyle\bigoplus_{i<j\in I}H_{cris}^{2}(X_{k,ij}/K_{0}) \\ \end{CD}

The first, second, fourth and fifth vertical arrows are the direct sums of the arrows appearing in the Gysin sequences:
\begin{equation*}
0\rightarrow H_{cris}^{1}(X_{k,i}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{1}(U_{i}/K_{0})\rightarrow H_{rig}^{0}(D_{i}/K_{0})\xrightarrow{\rho_{i}} H_{cris}^{2}(X_{k,i}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{2}(U_{i}/K_{0})
\end{equation*}
and
\begin{equation*}
0\rightarrow H_{cris}^{1}(X_{k,ij}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{1}(U_{ij}/K_{0})\rightarrow H_{rig}^{0}(D_{ij}/K_{0})\xrightarrow{\rho_{ij}} H_{cris}^{2}(X_{k,ij}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{2}(U_{ij}/K_{0})
\end{equation*}
We get a diagram

\begin{CD}
0 @>>> \mathrm{coker}(\alpha) @>>> H_{log-cris}^{2}(X_{k}/K_{0}) @>>> \ker(\beta) @>>> 0\\
@. @AAA @A\psi^{\ast}AA @AAA @.\\
0 @>>> \mathrm{coker}(\alpha') @>>> H_{rig}^{2}(X_{k}/K_{0}) @>>> \ker(\beta') @>>> 0 \\ \end{CD}

and from the Gysin sequences we can see that the left vertical arrow is an injection, but the kernel of the right vertical arrow will be
\begin{equation*}
\ker\left(\bigoplus_{i\in I}\rho_{i}(H_{rig}^{0}(D_{i}/K_{0})\rightarrow\bigoplus_{i<j\in I}\rho_{ij}(H_{rig}^{0}(D_{ij}/K_{0}))\right).
\end{equation*}
Probably this kernel is non-trivial in general, and hence $\psi^{\ast}:H_{rig}^{2}(X_{k}/K_{0})\rightarrow H_{log-cris}^{2}(X_{k}/K_{0})$ won't be injective in general. The same argument should work for all $n\geq 2$.

"Rigid cohomology and invariant cycles for a semistable log scheme". He shows that the weight-monodromy conjecture implies that there is an exact sequence $H_{rig}^{i}(X_{k}/K_{0})\rightarrow H_{log-cris}^{i}(X_{k}/K_{0})\xrightarrow{N}H_{log-cris}^{i}(X_{k}/K_{0})$. This is unconditional if, for example, $X$ is a semistable family of curves or surfaces. In these cases the first map is an injection for $i=1$. For searching purposes, the elements of $\ker N$ are often called "$p$-adic invariant cycles". $\endgroup$ – Oli Gregory Apr 10 at 13:28