I wanted to add some things to the comments I had already made but the list of comments have become very large and the comments I have already made are becoming more and more difficult to follow so I'll put everything (including the things I have already said) here instead even though it is not an answer to the question.

Let us first consider the case of a minimal surface $X$ (by minimal I mean $K_X$ nef). Dolgachev (Dolgachev: Reflection groups in algebraic geometry is a good reference even though the proof is only referenced there not given) gave a kind of structure theorem for the image $A_X$ of $\mathrm{Aut}(X)$ in $\mathrm{Aut}(S_X)$, where $S_X$ is the orthogonal complement of $K_X$ in $\mathrm{NS}(X)$ modulo torsion. His result says that there is a normal subgroup $W_X$ of $\mathrm{Aut}(S_X)$ generated by reflections in $-2$-curves and the group $P_X$ generated by $A_X$ and $W_X$ is a semi-direct product and of finite index in $\mathrm{Aut}(S_X)$. Note that it is possible to have $W_X=\{e\}$ and then $A_X$ itself is of finite index and hence an arithmetic (and thus finitely presented). It is also possible to have $W_X$ of finite index and then $A_X$ is finite (and thus finitely presented). However, there are intermediate cases where both $A_X$ and $W_X$ are infinite. Still $A_X$ is a quotient of $P_X$ and hence is finitely generated. I do not know if it is always finitely presented. Borcherds (Coxeter groups, Lorentzian lattices, and $K3$ surfaces. Internat. Math. Res. Notices 1998) gives examples where it is (and where it is even nicer) but also examples where it is finitely generated but not arithmetic.

A further step would be to blow up points of $X$ (still assumed minimal). As $X$ is the unique minimal model any automorphism of the blowing up is given by an automorphism of $X$ that permutes the blown up points (and the subgroup fixing the points is commensurable with the full automorphism group). In the case of abelian or hyperelliptic surfaces blowing up just one point is pointless as it just serves to kill off the connected component of $\mathrm{Aut}(X)$ so in that case the first interesting case is blowing up two points.

Consider the case of blowing up two points when $X$ is abelian. So we have two points on $X$ one of which we can assume is $0$ and the other we'll call $x$. An automorphism of $X$ that fixes both of these points will be en automorphism of $X$ as abelian variety that fixes pointwise the closed subgroup $A$ generated by $x$. The group fixing $x$ will then have finite index in the the group fixing $A$ pointwise. For any abelian subvariety $A$ of $X$, the subgroup of $\mathrm{Aut}(X)$ fixing all the points of $A$ is an arithmetic subgroup (in a not necessarily semi-simple group) and in particular is finitely presented.

The same argument works for abelian varieties of any dimension. There one of course also has the option of blowing up positive dimensional varieties, assume $S$ is a smooth closed subvariety. This time the automorphism group is the subgroup of automorphisms $X$ that fixes $S$. We thus get an induced action on $S$ and the kernel of that action has the same structure as before. Unless I am mistaken, the automorphisms of $S$ that extend to $X$ are of finite index in $\mathrm{Aut}(S)$ (look at $\mathrm{Alb}(S) \rightarrow X$ and split it up to isogeny). Hence the finite generation etc for the blowing up is reduced to finite generation for $S$ (and conversely for $X$ replaced by $\mathrm{Alb}(S)$).

Consider now the case of $X$ still minimal but non-abelian or hyper-elliptic and look at blowing up of one point $x$. For a general point of $X$ (in the sense of being outside a countable number of proper subvarieties) the automorphism group is trivial and hence finitely generated. The situation for arbitrary $x$ seems unclear but one thread of the discussion started concerning itself with whether for a general $X$ there is a characterisation (up to commensurability) of $\mathrm{Aut}(X)$ similar to the minimal case: Look at all automorphisms of the integral cohomology of $X$ that preserves multiplicative structure, Hodge structure, Chern classes (of the tangent bundle) and effective cones (spanned by effective cycles). Is the image of the automorphism group of $X$ of finite index in this group? I think the answer is no (and I hope that what I present here is a proof). For that we need to recall some facts on Seshadri constants (Lazarsfeld: Positivity in Algebraic Geometry, I is my reference). Given a point $x$ the Seshadri constants $\epsilon(L;x)$ for $L$ nef (but also for $L$ restricted to be ample) determine (and are determined by) the nef cone of the blowing up at $x$; $L-rE$ is nef precisely when $0\leq r\leq \epsilon(L;x)$. Switching tack, there is a subset $U$ of $X$ which is the intersection of a countable number of open non-empty subsets of $X$ such that $\epsilon(L;x)$ is constant on $U$ for all ample $L$. Indeed, $\epsilon(L;x)$ can be expressed (loc. cit.: 5.1.17) in terms of whether or not $kL$ separates $s$-jets at $x$ and for fixed $k$ and $s$ the separation is true on an open subset.

The conclusion is that there is a $U$ which is the intersection of a countable number of non-empty open subsets for which the nef cone of the blowing up of $X$ at $x$ is independent of $x$ when one expresses it in the decomposition $\mathrm{NS}(X)\bigoplus\mathrm Z E$. If we assume now that $K_X$ is numerically trivial we have that the first Chern class of the tangent bundle of the blowup of $X$ at some $x$ equals $E$ (up to torsion) and hence the group above will preserve the decomposition $\mathrm{NS}(X)\bigoplus\mathrm Z E$ and fix $E$ so come from an automorphism of $\mathrm{NS}(X)$. The only further condition we put on it is that it preserve the nef cone but for $x\in U$ this cone is independent of $x$. As we can further arrange it so that $x\in U \implies \varphi(x)\in U$ (as $\mathrm{Aut}(X)$ is countable) we get that all elements of $\mathrm{Aut}(X)$ give structure preserving automorphisms of the cohomology of the blowup of $X$ at $x$. However, as observed before, at the price of shrinking $U$ we can assume that that the automorphism group of the blowing up is trivial. Hence, if we let $X$ be for instance a K3-surface with infinite automorphism group we get an example.