The type 38 was adopted in the 38th year of the Japanese empire and the type 99 in the 99th year of the empire. So the 99 came much later...
while the Japanese dated many things for the year of introduction, using a system based on the year of the Emperor's reign (which reset with each new Emperor) it was not their only system.
The type 99 did come well after the Type 38, but not as long as the numbers suggest.
The Type 38, and the 6.5mm round were introduced in 1905. This lines up with the 38th year of the Meiji period (1868-1912). The Taisho period followed (1912-1926) and then the Showa period (1926-1989)
The Type 99 was introduced in 1939. This was not the 99th year of any Emperor, they used the Imperial year (2599) instead. The most famous example of this dating system is the Zero Fighter. The Mitsubishi A6M fighter plane was adopted by the Japanese Navy in 1940. This was Imperial year 2600 , and the plane was designated "Type 0", commonly known as the Zero thereafter.
Before you buy any ammo for your Arisaka, check the BORE, at the muzzle, to see if it is a 6.5mm (.26cal) or 7.7mm (.30 cal). easiest is to use a .30 bullet. Stick it in the muzzle. If its a 6.5 the .30 bullet won't go very far in. If its a 7.7 the .30 bullet will go in nearly up to its widest point.
MANY Arisakas were rebored in the US, since Japanese ammo was unobtainable for many years, bores and or chambers were often recut, to take something the user could get, either another commercial round or a wildcat. Sometimes without being marked as such.
The best way to be certain is to take the rifle to a gunsmith, and have the bore measured and a chamber cast done.
This will tell you what the rifle you have actually is, which might be different from what it ought to be, based on its model.