My absolute t-value is greater than t-critical value. This means that I can reject my null hypothesis which was that $\beta_1\leq 0$. Therefore, $\beta_1\gt 0$ and my alternative hypothesis is correct. However, my data has a negative slope. which tells me that $\beta_1\lt 0$. What am I missing here?
You seem to have $H_0: \beta_1 \le 0$ and $H_1: \beta_1 > 0$. In this case, you reject $H_0$ in favor of $H_1$ if your t-value (not the absolute t-value, but the t-value itself) is bigger than the one-tailed critical value. In your case, $\hat\beta_1 < 0$ so your t-value is negative and so you do not reject $H_0$ in favor of $H_1$.
Regarding significant* variables with the wrong signs (negative when expecting positive in your case), Jeffrey Wooldridge (Introductory Econometrics: A Modern Approach, 7e, end of Section 4-2f, p. 134) writes:
"A significant variable that has the unexpected sign and a practically large effect is much more troubling and difficult to resolve. One must usually think more about the model and the nature of the data to solve such problems. Often, a counterintuitive, significant estimate results from the omission of a key variable or from one of the important problems we will discuss in Chapters 9 and 15."
Note *: "Significance" means rejection of the null in favor of the two-sided alternative. "Significance" does not necessary imply the rejection of the null in favor of a one-sided alternative.