Using a priority queue is almost certainly the simplest solution, and also the most practical one, since it's O(n) storage (with a log factor if you require bignums). Any solution which involves computing all possible sums and putting them in a map will require O(n^2) storage, which soon becomes impractical.
My naive, non-optimized implementation using a priority queue is O(n^2 log(n)) time. Even so, it took less than five seconds for n = 10000 and about 750 seconds for n = 100000, using a couple of megabytes of storage. It certainly could be improved.
The basic idea, as per your comment, is to initialize a priority queue with pairs (a, a+1) for a in the range [1, N), and then repeatedly increment the second value of the smallest (by sum of cubes) tuple until it reaches N. If at any time the smallest two elements in the queue are equal, you have a solution. (I could paste the code, but you only asked for a hint.)