There are three components to your question, and I will try to answer them separately.
Regarding storing a multi-valued mapping, while there are specialized solutions available, the most common recommendation is just to store a hash whose values are arrays. That is, for your use case, your primary data structure is a hash whose keys are strings and whose values are arrays of integers. Depending on your desired behavior for duplicates etc., etc, you may wish to substitute a different data structure for the value structure, possibly a set.
Regarding identifying strings containing numbers and strings not containing numbers, well, that depends on exactly what your non-number-containing strings could instead contain, but a good starting point would be to perform a regular expression match for digits. You didn't specify whether your allowable numeric strings represented integers, floating points, etc. The particular answer to that may affect your overall strategy. Unfortunately, input parsing and validation is a complex and messy topic in the general case.
Regarding the actual conversion process, I would recommend the following strategy. Iterate through your input array. Check each string for whether it is numeric or non-numeric. If it is non-numeric, store that as the current key in a local. Also, in your hash, create a mapping from that key to a new empty array. If, instead, the string is numeric, convert it into a number, and add it to the array under the appropriate key.