Understanding Zechariah's prophecy in Luke 1 as a Protestant, Evangelical Dispensationalist believer In Luke 1:68–69, Zechariah says under the influence of the Holy Spirit:


  “Blessed be the Lord God of Israel,
      for he has visited and redeemed his people
   and has raised up a horn of salvation for us
      in the house of his servant David,"


ESV uses past perfect tense: "has visited and redeemed" as well as "has raised up."

From the context of the rest of his prophetic prayer of praise, some might infer that Zechariah refers here only to God saving physically, by His presence or intervention in the past.  

God was silent for about 400 years before sending Gabriel to Zechariah and then Mary.

Did Zechariah grasp the spiritual implication of the salvation that was at hand from reading "the prophets from of old" (70) through whom God had spoken?  

He speaks of promises in regard to  being "saved from our enemies and from the hand of all who hate us;" (71)

But this had a spiritual purpose:


  "that we...might serve him without fear,
  in holiness and righteousness before him all our days." (74–75)


He also says that his son, John,


  will go before the Lord to prepare his ways,
   to give knowledge of salvation to his people
      &nbsp;&nbsp;&nbsp;&nbsp;in the forgiveness of their sins,
   because of the tender mercy of our God,
      &nbsp;&nbsp;&nbsp;&nbsp;whereby the sunrise shall visit us from on high
   to give light to those who sit in darkness and in the shadow of death,
      &nbsp;&nbsp;&nbsp;&nbsp;to guide our feet into the way of peace.”


Zechariah had nine months to study what Gabriel had said and what was recorded by the prophets of old.  

1) What did he understand about spiritual salvation and the Lord's visitation through the child to be born to Mary?  (He likely heard her account of Gabriel's visit to her during the three months that she stayed with them.)  

2) What specific prophesies in the Old Testament, and fulfillments during Jesus' first advent, can be connected with Zechariah's prophecy?
The dispensational Moody Bible Commentary seems to address both of your questions at a high level.  On this passage, it reads:


  With his restored powers of speech, and being filled with the Holy Spirit [...], Zacharias immediately turned to praising God.  This song—a work of poetry—was intended to be the answer to the people's question in 1:66—What will this child turn out to be?  The song, drawn mostly from quotations and allusions of the OT (e.g., 1:68 with 1Kg 1:48; Ps 41:13; 72:18; 1:69 with 1Sm 2:1, 10; Ps 18:2; 89:17; 1:71 with Ps 106:10; 1:73 with Gn 22:16; 1:78 with Mal 4:2; 1:79 with Is 9:2), is called the Benedictus (after the first word of Lk 1:68 in the Latin version).  In one long sentence (1:68–75) Zacharias praised God for His work of fulfilling His covenant promises to David (1:69) and to Abraham (1:73).  This work of "covenant fulfillment" was to be accomplished by the One whom John would one day announce and introduce to the world—this child (you, child 1:76).  In the rest of the song (1:77–79) Zacharias highlighted the ministry of the coming One in terms of salvation, forgiveness, mercy, guidance, and peace. [emphasis in original]


So we see that Moody does indicate that Zacharias understands the idea of spiritual salvation that "was to be accomplished by the One whom John would one day announce," particularly on the basis of verses 76–79, and it links the Benedictus to many OT passages.  Most of these are allusions and not necessarily prophecies, but perhaps 1 Samuel 2:10 and especially Isaiah 9:2 would be understood as Messianic prophecies.  And Moody sees Zacharias's prophecy being fulfilled in general terms through the ministry of Jesus, "in terms of salvation, forgiveness, mercy, guidance, and peace."

Dispensationalist theologian Thomas Constable goes into more detail in his commentary on Luke.  He connects the "horn of salvation" of 1:69 with Ps 18:2, and argues that Zechariah is speaking of Jesus, knowing that the birth of Jesus was imminent because of Mary's visit. He also believes that the reference to "Lord" in 1:76 is primarily a reference to the Messiah (not simply God).  And he finds a reference to Numbers 24:17 ("a star shall come out of Jacob") in the phrase "sunrise shall visit us" of 1:78, and he too connects 1:79 with Isaiah 9:2.  

Like Moody, Constable doesn't connect specific events in Jesus' ministry to the prophecy of Zechariah, but clearly sees a connection between the prophecy and the Messiah's salvation (both political and spiritual) and the fulfillment of the covenant promises.