RSA uses internal padding, so you don't need to pad the input. RSA encryption can normally be performed in two different padding modes: OAEP and PKCS#1.5 padding. Both add substantial padding to each block. This is different from e.g. AES padding, where the padding is only performed on the last block to be encrypted. Also different is that the security of RSA heavily relies on the padding, so it must be performed.
Now, because each block is padded, obviously the plain text that fits into the block is less then the key size (the size of the modulus). In other words, the block size is smaller than the key size. In the case of PKCS#1.5 it is 11 bytes less than the key size (if this is on a byte boundary at least). Keeping to a slightly higher (say up to 19 bytes) block size would be better regarding security though, 8 bytes of the padding should be (secure) random bytes.
With Java you can actually ask the block size from the Cipher
object, and for RSA it will correctly subtract the overhead from the key size. Otherwise you will have to calculate it yourself. As said, you don't require padding of the last block, the block padding will fill the last block for you.
Note that the RSA PKCS#1 specifications are a free download. So you can do the calculations yourself for the (tougher) OAEP padding. For your particular assignment I would keep to 1.5 padding though.