Are t-tests good choices here? I would like to test if two groups of participants (dark-eyed and light-eyed individuals)  respond to a drug differently over time. The hypothesis is that the drug reaches its peak effects sooner but diminishes quicker in light-eyed individuals. Specifically, the following hypotheses will be tested:

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*light-eyed individuals have on average a higher biomarker level for the drug 1 hour after receiving the drug.

*light-eyed individuals have on average a higher biomarker level for the drug 2 hours after receiving the drug.

*light-eyed individuals have on average a lower biomarker level for the drug 4 hours after receiving the drug.

*light-eyed individuals have on average a lower biomarker level for the drug 8 hours after receiving the drug (the primary hypothesis).

*light-eyed individuals have on average a lower biomarker level for the drug 24 hours after receiving the drug.

Are doing 5 individual t-tests appropriate for the hypotheses above? Assuming there is no significant outlier, the sample mean will be a good summary stat to test if the drug levels in both groups are statistically significantly different at a specific time point.
For the normality assumption, the sample size will be 23 for each group (based on a power analysis using the 8 hours mark), so the CLT will likely kick in. I will check the histograms to make sure after the data is collected. The independence assumption will also hold since the measurement for one participant doesn't affect the other.
I am aware of the dependence between the hypotheses - if the drug level is lower at 4 hours for the light-eyed group, then it's likely to be also lower at 8 hours. Therefore, I will do a Bonferroni correction to adjust the alpha level.
Are there any non-parametric alternatives that would be better suited to testing those hypotheses?

Note that the CLT does not "kick in".  It moves at a glacial pace in many cases.
You did not fully specify the design so I assume it is a longitudinal one with multiple outcome measurements per subject.  It is important to correctly model the intra-subject correlation pattern.  A serial correlation model has the highest probability of fitting.  The most general way to fit such a model is with a simple Markov process.
Instead of making contrasts at a bunch of somewhat arbitrary time points, consider modeling time continuously (without assuming linearity, e.g., using a restricted cubic spline function) and estimating the difference in time-response profiles simultaneously across all times.  If you want to take this to the next level, compute simultaneous confidence bands for these differences to get an efficient multiplicity adjustment.
To get a robust analysis without assuming normality or assuming you have properly transformed $Y$, consider a semiparametric model such as the proportional odds model, which generalizes Wilcoxon-Mann-Whitney-Kruskal-Wallis tests.  The easiest ways to convert such models to longitudinal ones are Markov processes and random effects.  For the Markov approach a detailed case study is in the last chapter of https://hbiostat.org/rmsc.
