The phrases you need are "price per unit/item" or "price for each unit/item". We don't use "price for unit"... that's broken English.

For can be used universally. The price for something. Per implies that buying more than one is an option or that people may quite often purchase more than one. An example would be you would not tell a person looking at buy an airplane that the cost per airplane is X unless you knew that person was buying more than one. However, a company might buy a fleet and so the price per airplane would be appropriate. You could also say the price "for each" would have roughly the same meaning as "per." Another way per is used is with phone minutes and messages. Since someone would normally talk for more than 1 minute and send more than 1 message, you will more often than not see a per minute or per message breakdown of the cost.

You would normally use 'for' when items are sold in batch. For example: 5 oranges for $2. 3 apples for $1 2 movie tickets for $20 Conversely, if you are dealing with individual units, you would then use the word 'per'. For example: $3 per 100g of smoked beef $4 per ounce of liquid