What was the "invariable measure of value" problem and how did Sraffa solve it?

The working paper by Kurose & Yoshihara (2014) addresses both of your questions.

What was the "invariable measure of value" problem:

According to the Kurose & Yoshihara:

An invariable measure of value can be defined as a measure that is invariable with respect to changes in both income distribution and technique (Ricardo, 1951A, chap. 1). The advantage of the invariable measure of value, if it exists, is that we can distinguish between the variations which belong to the commodity itself and those which are occasioned by a variation in the medium by which values or prices are expressed, when relative prices change (Ricardo, 1951A, p. 48).

The invariable measure of value problem is the problem of finding such measure. Ricardo considered gold/silver money as good although flawed approximation. The authors provide more detailed overview of the idea at the start of chapter 2.

How did Sraffa 'solve it':

I put the '' around solve it as whether he indeed solved the issue is somewhat up to the debate (see the paper sec. 2.3), but the Sraffa's solution was as as follows (Kurose & Yoshihara, 2014):

... Sraffa (1960) revived the concern about the invariable measure of value, which had fallen into oblivion since the so-called Marginal Revolution. Unlike Ricardo, he divided the problem of identifying an invariable measure of value into two parts: the first is to search for a measure of value that is invariable with respect to changes in technique, left aside the change in income distribution, and the other is to search for a measure of value that is invariable with respect to the change in income distribution, left aside the change in technique. Sraffa concentrated on the latter by constructing a special, composite commodity termed the standard commodity. He also demonstrated an interesting relationship with respect to income distribution if the standard commodity is adopted as the numéraire: the linear relationship of income distribution.

Again you will find more detailed overview of Sraffa's solution in the paper as well as generalization of said solution by the authors.