Tax system is progressive or regressive
Suppose you learn that the richest 10% of people pay 30% of income
  tax. Does it mean that the tax system is progressive?


You are right. It cannot be determined without knowing the income tax of lower-income earners.


  If there are no eligibility restrictions, is this policy likely to be
  progressive or regressive? What might justify this policy?


That is even more uncertain than the previous question because it requires assumptions that are not inherent to --or implied by-- the policy. In fact, giving scholarships to some of the best students contradicts that there are no eligibility restrictions, but that is irrelevant.

First, not all degrees (whether earned at home or abroad) lead to a profitable career. For instance, graduates in graphical design usually earn significantly less than physicians.

Second, the effect of scholarships also depends on whether those students become high-income earners and decide sever ties with their home country. This could lead to a "progressive" scenario of lower inequality notwithstanding that the population that remains in the home country is the poorer sector.

It is just impossible to conjecture about the effects of the policy based on the information given.