There is currently no object analog to the $arrayElemAt expression that would allow you to easily access an object value at a variable key returned by an expression. Because of this, the most direct way to achieve this in a single step is to do the following:

{
    $addFields: {
        valueAtKey: {
            $let: {
                vars: { keyVals: { $objectToArray: '$someObjectField' } },
                in: {
                    $arrayElemAt: [
                        '$$keyVals',
                         { $indexOfArray: ['$$keyVals.k', '$someKeyField'] }
                    ]
                }
            }
        }
    }
}

This seems really roundabout and, on top of that, it actually returns the { k: ..., v: ... } object, so either another nested $let or a following $addFields stage would be needed if you just wanted the value.

It would be much more ideal, if possible, to be able to do something similar syntax-wise to $arrayElemAt like the following:

{
    $addFields: {
        valueAtKey: { $objectValueAt: ['$someObjectField', '$someKeyField'] }
    }
}

This would, ideally, resolve directly to the value if found and null otherwise. There would probably also need to be an error case, similar to $arrayElemAt, when the second argument does not resolve to a string.