What type of ANOVA test should I do?

The same participants are exposed to two different screen sizes 27" (stimulus A) and screen size 9.7" (stimulus B). Independent var is "screen" with two levels - sizes. After exposure to stimulus A, they fill two questionnaires to answer dependent variables Y1 and Y2. After that, they are exposed to stimulus B, and asked to fill the same above questionnaires to respond to the same dependent var Y1 and Y2.

Y1: Attention Y2: Endurability

I want to compare each set of their responses (dependent var) of each stimulus. What kind of ANOVA test should I conduct?
This sounds like something for a repeated-measures MANOVA. Your within-subject factor is the stimulus (A vs. B) and the dependent variable is a vector (Y1, Y2) of responses.

Alternatively, you could perform a separate repeated-measures ANOVA (or equivalently, a paired t-test) on each dependent variable and do a Bonferroni adjustment to control the family-wise error rate, if desired. However, this would not incorporate information on the possible correlation between Y1 and Y2.
Thank you for the kind reply. Just to clarify, are you referring to a one-way within subject MANOVA test? Since we have only 1 independent var with 2 levels, we conduct a on-way within subject test, and as there are two response var Y1 and Y2, we conduct a MANOVA (to minimize Type I errors).

There it is not necessary to conduct a regression analysis as a MONOVA will provide us with the correlation in the output (i am presuming r-square).

Kindly confirm.
Yes. I don't believe MANOVA output will provide correlations among the dependent variables, but some may provide partial eta squared, which is analogous to R-square.