Is Solas run by ETB?

No, in fact it's the other way around. SOLAS (An tSeirbhís Oideachais Leanúnaigh agus Scileanna) is a state agency in Ireland responsible for the Further Education and Training (FET) sector, while ETBs are the bodies that deliver the training programs. SOLAS was established by the Department of Education and Skills (now the Department of Further and Higher Education, Research, Innovation and Science) to oversee and fund the provision of Further Education in Ireland. The ETBs, are statutory bodies which manage and operate Community National Schools, Post-Primary Schools, Further Education colleges, and various adult and further education centres in Ireland.

SOLAS (An tSeirbhís Oideachais Leanúnaigh agus Scileanna):

A state agency established in 2013.
Operates under the Department of Further and Higher Education, Research, Innovation and Science.
Responsible for overseeing the Further Education and Training (FET) sector in Ireland.
Funds and develops national FET strategies and policies.

Education and Training Boards (ETBs):

Regional bodies responsible for the delivery of FET programs within their respective areas (as well as primary, post-primary and adult education).
Work with SOLAS to implement national strategies and courses.
Provide administrative and operational support for training centres.
Essentially, SOLAS sets the direction of Further Education and provides funding, while the ETBs implement the courses and deliver the training on a regional level.