Can a sphere with dimension $2$ admit locally negative curvature？ I know that by the Cartan-Hadamard theorem the sphere cannot admit a global negative curvature, which can also be proved by the Gauss-Bonnet formula.
But can we equip the sphere with a metric such that its curvature is negative at some point？
 A: Several constructions have been mentioned in the comments. Here's another method to produce such metrics which generalises to other situations.
Let $g$ be a Riemannian metric on an $n$-dimensional manifold, and let $\tilde{g} = e^{2\varphi}g$ be a conformal metric. Then their scalar curvatures are related by the formula
$$s_{\tilde{g}} = e^{-2\varphi}\left[s_g + 2(n-1)\Delta_g\varphi - (n-2)(n-1)|d\varphi|^2\right]$$
where $\Delta_g = d^*d + dd^*$ is the Laplace-de Rham operator (which is the negative of the Laplace-Beltrami operator). In particular, when $n = 2$, we have
$$s_{\tilde{g}} = e^{-2\varphi}\left[s_g + 2\Delta_g\varphi\right].$$
On $S^2$, we can take $g$ to be the round metric, in which case $s_g \equiv 1$, so $s_{\tilde{g}} = e^{-2\varphi}[1+2\Delta_g\varphi]$. Therefore, if $\varphi$ is a real-valued function on $S^2$ with $(\Delta_g\varphi)(p) < -\frac{1}{2}$, then $s_{\tilde{g}}(p) < 0$. Moreover, by the continuity of $s_{\tilde{g}}$, the scalar curvature of $\tilde{g}$ is negative in an open neighbourhood of $p$.
Your question can be regarded as a special case of the prescribed scalar curvature problem:

Which functions $s \in C^{\infty}(M)$ arise as the scalar curvature of some Riemannian metric $g$ on $M$?

As you alluded to in your question, if $s \in C^{\infty}(S^2)$ is the scalar curvature of some metric, then $s$ must be positive somewhere due to the Gauss-Bonnet Theorem. Kazdan and Warner showed that this necessary condition is actually sufficient! That is, a function $s \in C^{\infty}(S^2)$ is the scalar curvature of some Riemannian metric $g$ on $S^2$ if and only if $s$ is positive somewhere.
Kazdan and Warner solved the prescribed scalar curvature problem for all closed two-dimensional manifolds. Depending on the sign of the Euler characteristic, a necessary condition can be derived from the Gauss-Bonnet Theorem as we did above, and this condition is sufficient. Moreover, you can choose the metric to be conformal to any fixed constant scalar curvature metric, see Theorem 5.1 of Existence and Conformal Deformation of Metrics With Prescribed Gaussian and Scalar Curvatures.
Kazdan and Warner also solved the prescribed scalar curvature problem for higher-dimensional closed manifolds. See Theorem 7.11 of these notes by Kazdan for example.
