What ideological and cultural challenges does democracy promotion face?
The answer to this varies depending on the country or region in which one is trying to promote democracy. The problems of promoting democracy in, for example, Burma are different from the problems of promoting it in Egypt.
In terms of culture, there are cultures that do not promote the idea of individualism and personal rights. Asian countries, for example, often speak of Confucian ideals and the need to subordinate personal desires to the needs of the larger community. This is an idea that clearly goes against the ideas of democracy. Some Muslims similarly believe in the need for religious authorities to have power in a country. This, too, can be a challenge to the promotion of democracy.
There are fewer ideological challenges today as the Cold War recedes into the past. Challenges to democracy tend to come more from religion, culture, or simply the desire of the powerful to hold on to their power. However, if we say that authoritarianism is an ideology, that is an ideological challenge to democracy that can be seen in many places. Russia is a good example of this as its people generally believe (see economist.com link below) that freedom from governmental interference is a much less important thing than material progress. This sort of ideological attitude makes it harder to spread democracy.