Just because Art and Culture are essential to a society does not mean they should have a ministry. It is not a disrespect to the concept of culture to not regulate it. When it comes to the creation of a ministry, it is a question of politics, not respect. Politically, we do not need a ministry of Arts and Culture and I will argue that having one would actually be a bad thing.
First, we have to ask, what will a ministry do? Primarily, it will regulate. This means it will enforce rules. It will restrict. It will limit. It will have the potential to censor. I imagine most artists would not consider this a good thing. The only positive I can see here is copy writing and patents, however, this does not require the resources of an entire ministry. Also, copy write laws are arguably overabundant and sometimes harmful currently, so it's unlikely creating a larger and more complicated system than ones currently used in many countries would benefit us. Regardless, this only really applies to Art.
So what is culture? Google defines it as "the arts and other manifestations of human intellectual achievement regarded collectively." So culture is the collective manifestations of human achievement. In other words, our culture is just what we naturally do over time as a collective, This can be directly influenced by a government of course, but only in the manner of mass manipulation. Is that what we want from our ministry of culture? What else would it do? The regulation of culture is nothing but censorship. If we are hoping for the preservation of culture, then we are devoting an entire ministry to the management of a museum, an incredible inefficiency. Lastly, an official system for judging the merit of a cultural or artistic creation would be a dangerous monster. There are few things more dangerous then a political institution with the authority to tell it's citizens which thoughts are better than others.
Art and culture will happen on their own. We cannot artificially create them through a ministry. They do not need regulation. Of course they are important, but importance does not create political need. It is the job of individual Asgardians to manage our culture collectively. It will happen on it's own. A ministry is a very large use of resources with the purpose of regulating something. Art and culture are supremely important, but they do not need regulation.