It is possible to get the variable context for the plan from the following REST:
<Bamboo_URL>/rest/api/latest/plan/<PLAN>?expand=variableContext

This returns the variables as they are configured right at the moment we call the REST, since this is the plan configuration.

Unfortunately there is no way to get the variable from a specific build:
<Bamboo_URL>/rest/api/latest/result/<BUILD-KEY>?expand=x.x.variableContext