Who was the original service provider for the Alcatel One Touch" and is that phone 'unlocked'. If the original provider was on a different network type, then no, you couldn't use it. The same goes for if the phone is locked.
Your best bet, and to know for sure, call "Straight Talk" and ask them if that specific phone can be used on their network. You don't have to give them any personal info to ask, so you don't have to worry about them messing with your service. Just ask the CSR you speak to if Straight Talk will work on that specific Alcatel phone (there are many versions of 'one touch' so you will need the full phone model). You may also need to tell them the original service provider for that Alcatel phone, so they will know any...