Translation:He is running into the castle.
Couldn't this also be translated "he RAN into the castle"? How would the Hebrew be different
Yes, both are correct. No way to tell the difference other than the context.
In fact running in should be בתוך not לתוך
בתוך = Inside
לתוך = Into
What is the difference between טירה and ארמון?
I think they are nearly the same. Maby טִירָה is a bit more of a "castle", but אַרְמוֹן more a "palace"
That depends on whether the gate is open.