Here is a method which is very efficient in the case were "constructive" is interpreted as "no axiom of choice at all, not even countable and no law of excluded middle", i.e. essentially "topos logic".
It is possible to construct a very well behaved "Zariski spectrum" (including its structural sheaf whose globale section will be $A$ exactly) as a locale instead of a topological space: on can simply say that $\mathcal{O}(\text{Spec }A) := \{ \text{ radical ideal of } A \}$ or better, that $\text{Spec } A$ is the classifying space of the theory of "complement of prime ideals" described below. One can then construct the structural sheaf and so one and it is relatively trivial that the set of global section is $A$ and that an element of $A$ which is nowhere invertible in the structural sheaf is nilpotent.
points of the spectrum are still the prime ideal, but because it is now a locale instead of a topological space one does not really care about existence of points or not.
well this is not entirely true: technically the point (in the sense of classyfing topos) are the "complement of prime ideal" I.e. subset $I$ that satisfies $0 \notin I$, $1 \in I$, if $x+y \in I$ then $x\in I$ or $y \in I$ and $yx \in I$ if and only if $x \in I$ and $y \in I$. assuming the law of excluded middle this is the same as saying that the complement of $I$ is a prime ideal...
almost all "geometrical argument" can be made constructive by replacing the ordinary zariski spectrum by the localic Zariski spectrum, and this include most prof that involve using all prime ideal.
This technique is very well known among topos theorist but I don't know any reference explaining this clearly, maybe someone will know of one ?
In the mean time, I will try to give a little more explanation: Basically, instead of saying "let $\rho$ be a prime ideal" you move to the structural sheaf over the Zariski spectrum, especially if you know a little bit of internal logic this amount to assume that you have a subset $I$ as above which play the role of (the complement of) your prime ideal and if you can prove that your element $x$ is never in $I$ then it is nilpotent or if your some ideal "always contains an element of $I$" it has to be the whole ring and so one... Moreover, the structural sheaf is the localization at $I$ and $I$ is exactly the set of element that are invertible in the structural sheaf.
The drawback is that the rest of the proof has to be performed internally in in the topos of sheaves over the zariski spectrum, hence really has to be constructive (not involving the law of excluded middle) or has to involve working explicitly with sheaves.
Let me illustrate this on your two examples:
The sum of two nilpotent is nilpotent:
($I$ denote the universal "completment of prime ideal" in the logic of $spec A$, it is also the subobject of the structural sheaf of invertible element)
let $x$ and $y$ be nilpotent, internally in $\text{Spec } A$ $x$ and $y$ are not invertible (i.e. not element of $I$) hence $x+y$ is not invertible either (because $x+Y \in I \Rightarrow x \in I$ or $y \in Y$), hence $x+y$ is nowhere invertible on the spectrum hence nilpotent.
The thing about sum of ideals: well the exact same proof apply, just redefine $V(A)$ to be the corresponding closed subspace of the Zariski spectrum instead of a set of prime ideal...