The selection of one-sided or two sided test depends on what your objective is, what question you trying to answer. Most importantly, you should have set/defined those objectives before collecting your data [rather than contemplating what tests to use afterwards].
First of all, it is good that you are using a fisher exact test in this situation as you are in a small sample situation where the chi-squared approximation doesn't perform very well.
To delve a bit more into the details. If your objective is to detect the departure from the null hypothesis in either direction, then a two-sided test is appropriate. The null hypothesis is that the true Odds Ratio (OR)=1. If you want to be able to detect if your OR is <1 or >1, then a one-sided test would be appropriate.
Your two-sided test shows that the true OR is significantly less than 1 at 10% level of significance [but not at 5% level].
Your estimated OR is 0.13 and associated 95% CI is (0.00-1.23). As you can see your OR contains 1, hence, the significance is marginal at 5% level.
If your interest was to test that true odds ratio is <1 i.e. if you want to detect the departure from the null in only direction (one-tail only), then your test is significant at 5% level.
The OR is 0.13 and 95% CI is:0.00-0.95, which doesn't contain 1.