As you are an employee in the Republic of Ireland, your social insurance contributions are deducted by your employer in the normal way. The health contribution is also collected from your wages. You are entitled to NHS services because you are living in Northern Ireland therefore it is unlikely you will avail of health services in the South because of the cost.
With some exceptions National Insurance is not generally payable on earnings arising outside NI.  However, you may be entitled to pay voluntary National Insurance contributions to protect your UK state pension entitlement. Further information can be found at:  www.dsdni.gov.uk