I am Indian passport holder living in Hungary for 4 years and I have been issued a Temporary residence permit (EU Schengen).
Can I go to visit Dublin without any visa, or do I need to apply for a UK visa? I know Ireland still in Schengen but part of United Kingdom.
As noted before, Ireland is not part of the UK nor of Schengen. Though, Northern Ireland is part of the UK, but again, the UK is also not part of Schengen. Ireland and the UK have their own Common Travel Area, in which there are no permanent border controls between the two countries.
As an Indian citizen, you may be eligible for the British Irish Visa Scheme. Under this scheme, if you have a UK visa, you may use it to travel from the UK to Ireland, and if you have an Irish visa, you may use it to travel from Ireland to the UK. Anyone who is eligible and applies for one of these visas will generally receive a "BIVS" endorsement on their visa automatically, unless they are ineligible (e.g. they were previously removed from one of the two countries). If you do not get a BIVS endorsement, then you must also apply for a visa for the other country if you wish to travel to it.
The main thing to remember with the British Irish Visa Scheme is that you must always enter the Common Travel Area via the country which issued the visa. Only then can you travel to the other country. For instance, if you have an Irish visa, then on each journey you must always clear immigration in Ireland before going to the UK, and if you have a UK visa you must always clear immigration in the UK before going to Ireland. But you may exit the Common Travel Area via either country.
You can apply for an Irish visa by following the directions given by the Irish embassy in Hungary, or you can apply for a UK visa.
Ireland is neither a part of the UK nor in the Schengen area. Part of the island is indeed under UK sovereignty, while most of it is an independent country with its own visa policy. The UK and the Republic of Ireland are both part of something called the “Common Travel Area” (CTA), which means there are no border controls between the two countries but also that the Republic of Ireland cannot join the Schengen area, even if it wanted to.
The CTA is much less comprehensive than the Schengen agreement so while it does entail the creation of a border-free area, it did not create a common visa policy or a general right for UK visa holders to visit Ireland and vice versa (there are a few limited exceptions that could apply to you however).
Whether you need an Irish visa therefore depends on the rules detailed here. Indian citizens generally need a visa, even if you could in fact enter Ireland based on some UK visas (but you need to qualify for one of these visas in the first place). If your main goal is going to Ireland, the logical course of action is to apply for an Irish visa directly. I think they are cheaper too.
