I am living in Madrid, Spain, with a wife who is working here in Madrid too. I want to go to Ireland to visit as a tourist but without her. I have a Spanish residence permit of family member of the EU. Can I travel to Ireland without applying for Visa or what is it exactly I need to do? Do I need other travel documents to travel alone?
The general rule is that you can use your "family member of a Union citizen" Article 10 residence permit to travel without a visa, if you are traveling with or joining the EU citizen family member.
This is documented on Ireland's official web site:
Your EU/EEA or Swiss family member has rights under EU treaties that may allow you to accompany or join them in Ireland without a visa.
To exercise this right, you must apply for and hold a 'Residence card of a family member of a Union citizen' of the type referred to in Article 10 of Directive 2004/38/EC.
If you wish to travel alone, you will need to obtain a visa.