You probably have in mind a dynamical "Hamiltonian system", which is typically defined over a phase space (typically position-momentum). In this case it makes sense to talk about potential and kinetic energy.
Here you are doing equilibrium statistical mechanics. The $H$ of the Ising model is not the full Hamiltonian of the underlying dynamical system. In fact, at this basic level, the Ising model is not defined over a "phase space", but rather over a "configuration space" that is the space of all spin configurations. There is no explicit time (this is equilibrium thermodynamics/statistical mechanics), and there are no canonically conjugate variables.
From an abstract point of view (useful in optimization problems), the function $H$ of the Ising model may be called "cost of the configuration".
If you really want to introduce time, then you have to perform a Wick rotation (but by doing this you end up with a quantum system, see also this). In this case you will be able to "see" a discrete time derivative in the Hamiltonian, which can be used to define a sort of kinetic energy. However, this would not solve your conceptual problem, as the Wick rotation is just a formal way to map quantum systems into thermodynamic systems.
In other words: there is no way to compute the $i$-spin after a time $t$ given an initial configuration that specifies all the initial spins within the framework of equilibrium statistical mechanics. At equilibrium the temporal dynamics of each spin is ruled by thermal fluctuations, which description is beyond the formulation you are referring to in your question.
Edit: here I am not saying that every statistical mechanical model does not have a kinetic term. However, are some statistical mechanical models that are not defined on a phase space, but rather on a configuration space.
I also completely agree with the nice answer of @Qmechanic. You can add to $H$ a "kinetic" term $\sum_i^N \sigma_i ^2/2 = N/2$, which is just an irrelevant constant! However, this does not answer to the more "philosophical" part of OP's question. I believe that, in the end, the answer is: not all statistical mechanics models stem from an underlying microscopic Hamiltonian dynamical model (despite they have a so-called "Hamiltonian"). After all, the degrees of freedom appearing in the irrelevant kinetic term $N/2$ are still the $\sigma_i$, and not some canonically conjugate variables.