Besides Jesus, is there any other figure in the Bible that would claim that some of the prophecies in the Scriptures were about him?

Who says of Cyrus, ‘He is My shepherd, And he shall perform all My pleasure, Saying to Jerusalem, “You shall be built,” And to the temple, “Your foundation shall be laid.” ’ (Isaiah 44:28 NKJV)

Thus says the Lord to His anointed, To Cyrus, whose right hand I have held— To subdue nations before him And loose the armor of kings, To open before him the double doors, So that the gates will not be shut. (Isaiah 45:1 NKJV)

Isaiah's prophetic career, which began in 755 BC, was 200 years before Cyrus the Great became king of Persia in 559 BC. Cyrus was the king who freed the Jews and returned the exiles and plunder from Babylon to Jerusalem, fulfilling Isaiah's prophecy.

Alexander the Great. Not only does Daniel speak of his kingdom, but Job 20 prophesies his death:

Do you not know this from of old,
    since man was placed on earth,
5 that the exulting of the wicked is short,
    and the joy of the godless but for a moment?
6 Though his height mount up to the heavens,
    and his head reach to the clouds,
7 he will perish forever like his own dung;
    those who have seen him will say, ‘Where is he?’
8 He will fly away like a dream and not be found;
    he will be chased away like a vision of the night.
9 The eye that saw him will see him no more,
    nor will his place any more behold him.
10 His children will seek the favor of the poor,
    and his hands will give back his wealth.
11 His bones are full of his youthful vigor,
    but it will lie down with him in the dust.

12 “Though evil is sweet in his mouth,
    though he hides it under his tongue,
13 though he is loath to let it go
    and holds it in his mouth,
14 yet his food is turned in his stomach;
    it is the venom of cobras within him.
15 He swallows down riches and vomits them up again;
    God casts them out of his belly.
16 He will suck the poison of cobras;
    the tongue of a viper will kill him.
17 He will not look upon the rivers,
    the streams flowing with honey and curds.
18 He will give back the fruit of his toil
    and will not swallow it down;
from the profit of his trading
    he will get no enjoyment.
19 For he has crushed and abandoned the poor;
    he has seized a house that he did not build.

20 “Because he knew no contentment in his belly,
    he will not let anything in which he delights escape him.
21 There was nothing left after he had eaten;
    therefore his prosperity will not endure.
22 In the fullness of his sufficiency he will be in distress;
    the hand of everyone in misery will come against him.
23 To fill his belly to the full,
    God[a] will send his burning anger against him
    and rain it upon him into his body.
24 He will flee from an iron weapon;
    a bronze arrow will strike him through.
25 It is drawn forth and comes out of his body;
    the glittering point comes out of his gallbladder;
    terrors come upon him.
26 Utter darkness is laid up for his treasures;
    a fire not fanned will devour him;
    what is left in his tent will be consumed.
27 The heavens will reveal his iniquity,
    and the earth will rise up against him.
28 The possessions of his house will be carried away,
    dragged off in the day of God's[b] wrath.
29 This is the wicked man's portion from God,
    the heritage decreed for him by God.”

About twenty details of the death of Alexander the Great are found in this passage, including the circumstances of the start of his illness (at a party), his youth, most of his medical symptoms (abdominal and back pains, vomiting, fever, delirium, etc), the fate of his family (killed by one of his generals), and the fact that when buried, he would be encased in honey (which slows decay of corpses).

The below verses from the Gospel of John is where John the Baptist is quoting a prophecy about himself:

John 1:19-23 (NIV) Now this was John’s testimony when the Jewish leaders in Jerusalem sent priests and Levites to ask him who he was. He did not fail to confess, but confessed freely, “I am not the Messiah.”

They asked him, “Then who are you? Are you Elijah?”

He said, “I am not.”

“Are you the Prophet?”

He answered, “No.”

Finally they said, “Who are you? Give us an answer to take back to those who sent us. What do you say about yourself?” John replied in the words of Isaiah the prophet, “I am the voice of one calling in the wilderness, ‘Make straight the way for the Lord.’”

John the Baptist quoted from Isa 40:3:

Isa 40:3 (RSV) A voice cries: “In the wilderness prepare the way of the Lord, Make straight in the desert a highway for our God".

Isa 40:3 (NASB) A voice is calling, “Clear the way for the Lord in the wilderness; Make smooth in the desert a highway for our God".

Isa 40:3 (NKJV) The voice of one crying in the wilderness: “Prepare the way of the Lord; Make straight in the desert A highway for our God".

Isa 40:3 (TLB) Listen! I hear the voice of someone shouting, “Make a road for the Lord through the wilderness; make him a straight, smooth road through the desert".