A fair 6-sided die has three odd numbers: 1, 3, and 5. Therefore, the probability of rolling an odd number on a single roll is:
\[ P(\text{odd}) = \frac{\text{Number of odd outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]
When rolling the die 360 times, the expected number of times the event of landing on an odd number will occur can be calculated as follows:
\[ \text{Expected number of odd rolls} = \text{Total rolls} \times P(\text{odd}) = 360 \times \frac{1}{2} = 180 \]
Thus, a reasonable prediction for the number of times the event of landing on an odd number will occur is:
\[ \boxed{180} \]