I have very specific question regarding music. If I want to use some very old music (and only use the music without lyrics) from the song that is dated around first world war, or earlier that has a cult status in culture of some country, but the person holding the rights to the song is probably hard to find or unknown, Can I still get in problems with copyright or similar? Or this can fit in 'fair use'? Keep in mind this song and similar to this song are used in many movies and such.

The copyright on the composition and performance expire 75(?) years after the death of the composer and performer (the law may have changed recently).
So it depends on how old. You should consult your attorney.

Both the Berne Convention and the Universal Copyright Convention require 70 years after the author's death (used to be 50 years until ca. 1970), although individual countries are allowed to have longer terms. Mexico is an example (100 years), and there exist 2 or 3 countries in Africa which have 90-99 years. Almost all countries follow the minimum 70 years, however.

The Berne Copyright Convention is the entire planet except the USA, Iran, Iraq, Turkmenistan, Afghanistan, Angola, Ehtiopia, Somalia, Mozambique and 3 or 4 small countries in Asia.

The UCC includes pretty much all countries (all countries that could possibly matter, anyway).

If the author is not known, not precisely known, or if there are several authors, it's a lot more complicated because there exists no universal "international rule", and sometimes not even clear rules in the same country. For example, in the USA, it can be 95 years after publication or 120 years after creation. It's great when something can be either-or, any your money depends on it, and you can't tell for sure which one it is.

Also, do note that in some rogue countries, copyright expiration still does not mean that you can use something for free. For example here in Naziland, the GEMA has been pursuing christmas carolers for singing 16th and 19th century songs. Recently, kindergarten teachers are required to pay ransom for making photocopies of such songs, too.

O tempora, o mores!