To answer your question directly: No, there's no real mention of Zoroastrianism in the Bible, though Persians and Medes and other peoples who would've practiced Zoroastrianism or similar faiths certainly feature.

The influence of Zoroastrianism on Christianity comes from two main sources - one, Greek thought influenced by Zoroastrianism, and two, Hellenic Judaism influenced by the same, along with Achaemenid-era Judaism which was also influenced to a difficult-to-determine degree by Zoroastrianism and reactions against it, as can be seen in Isaiah. For influence on Judaism I have written a bit on this here.

Focusing more specifically on the elements that made it into Christianity:

• The most obvious one is messianism. All the way back in the Gathas composed around 1300 BC, Zoroaster speaks of "Saoshyant", "those who bring benefit", who will judge evildoers in the future. This was developed into what one might call a proto-messianic idea more fully attested in Middle Persian sources, turning the coming of the Saoshyant into the heralding of a final age, where good would be separated from evil again. The exact role of the Saoshyant varies, but includes for instance resurrection of the dead.

• Satan as a personification of evil. Though the Gospels still maintain remnants of the prosecutorial "ha-Shaitan" we know from Job ("And Jesus was driven by the spirit up into the wilderness to be tested by the Devil"), it appears that the dominant idea of the time was of Satan as an agent of evil, in direct opposition to God. It is hard here not to see the parallel to Angra Mainyu, the "Spirit of Destruction" who is the source of all evil in Zoroastrianism, in opposition to Spenta Mainyu, the "Holy Spirit".

• Arguably, some of the Neoplatonist-esque elements of Gnostic Christianity, with focus on mystical wisdom, might draw on Zoroastrian ideas by way of Greek thought.

• Bodily, in addition to spiritual, resurrection. An important feature of Zoroastrian burial rites was the collection of the bones, picked clean by scavengers, so that they would be resurrected along with the deceased person's spirit.

• God as a source of good, ethics, wisdom and all positive human qualities, as well as celebration specifically as a creator of the physical world. These are elements that are very important in hymns to Ahura Mazda, and seem to show up in Isaiah around the time Jews came into contact with Zoroastrians.

As for whether Zoroastrianism and Christianity are similar: I would say that especially ritually, Zoroastrianism has much more in common with Judaism (or Islam, even) than with Christianity, but certainly there are many common elements in their theologies.