How can I use time stamps for classification?
Not knowing your data in detail, my intuition is that you could go for dummy (one hot) encoding. You could split each day in (say) 10 min. intervals $x$ (144 columns) and attach labels (up/down) $y$. Each time interval $x$ would be one dummy encoded column (true/false).

The model would be a binary classification (logistic) like:

$$ y = \beta_0 + \beta_1 x_1 + ... + \beta_n x_n + u .$$

In case you have enough days, this could be a reasonable predictive model. It is likely, that some of the dummy encoded $x$ are irrelevant. So it could be worth to check a Lasso model, where some (not so helpful $x$) are shrunken or automatically excluded from the model. 

https://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html

Based on the same logic, you could also check boosting, e.g. lightGBM to make the classification.