It is not possible to apply for a PPS on behalf of someone else, the Polish person needs to apply for themselves and in order to do so, they must supply certain information. That is, various documentation proving their identity and residence in Ireland. The local social welfare office can give further information on how to apply and the various documents required.
It is not up to an employer to Register anyone for a PPS, an employee is obliged to hold a PPS Number and to supply it to an employer when commencing work. The fact that this person was employed without a PPS Number probably indicates to me that they were probably not registered as an employee so social insurance may not have been deducted from their pay. Either way, this person should apply for a PPS Number to their local social welfare office if they intend to remain here in Ireland.
The issue regarding the medical card is a different matter. Having a PPS number does not give anyone an automatic right to either a medical card or a GP Visit Card, but having a PPS Number is necessary in order to apply.
On the basis of living here for 4 weeks, it is unclear as to whether or not this person would even qualify as ordinarily resident for health services in Ireland as it appears from your post their wish/intention is to return home. If this person holds a Euroepan Health Insurance card issued by the Polish authorities, they can obtain necessary healthcare in Ireland.
If this person requires urgent medical attention, they can either present their Polish European Health Insurance Card to a GP that accepts public patients (your Local Health Office can supply a list) or a public hospital. If they do not have a European Health Insurance Card it is likely they will have to pay the full cost of treatment (that is, the cost of a GP visit and any in-patient or out-patient hospital charges).
It is not possible to provide further information on the rights of this individual without further information on their circumstances - PM me if you wish, but please include as much detail as you can.
CMCR.