How to perform data scaling/standardization on dataset containing grouped values?
There's not one right answer to this question, because what scaler works best really depends on the data and the algorithm you use to make the prediction. You should try different scalers combined with different algorithms to decide which preprocessing is best by comparing the cross validation results of each pipeline.
Of course, you don't have unlimited time. I would:

First look at the distribution of cost globally and see if you think you need to transform the target to make it more normal
Consider a tree-based algorithm for a first-pass which won't require you to scale your Problem_Size
Scale globally first, and then if you have time experiment with scaling based on problem type

You'll eventually run out of time or patience, but I think going in this order will make the most efficient use of your time.
Remember, there's not 1 best way to do it for every problem.