APL (Dyalog Unicode), 38 36 bytes

Thanks to ngn for fixing a bug while maintaining byte count.

Anonymous tacit prefix function. Requires ⎕IO (Index Origin) set to 0, which is default on many systems.

⍕,{2↑'thstndrd'↓⍨2×⊃⍵⌽∊1 0 8\⊂10↑⍳4}

Try it online!

{…} anonymous lambda; ⍵ is argument:

⍳4 first four ɩndices; [0,1,2,3]

10↑ take first ten elements from that, padding with zeros: [0,1,2,3,0,0,0,0,0,0]

⊂ enclose to treat as single element; [[0,1,2,3,0,0,0,0,0,0]]

1 0 8\ expand to one copy, a prototypical copy (all-zero), eight copies;
[[1,2,3,4,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0],
[1,2,3,4,0,0,0,0,0,0],
[1,2,3,4,0,0,0,0,0,0],
⋮ (5 more)
[1,2,3,4,0,0,0,0,0,0]]

∊ ϵnlist (flatten);
[1,2,3,4,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,
1,2,3,4,0,0,0,0,0,0,
1,2,3,4,0,0,0,0,0,0,
⋮ (50 more)
1,2,3,4,0,0,0,0,0,0]

⍵⌽ cyclically rotate left as many steps as indicated by the argument

⊃ pick the first number (i.e. the argument-mod-100'th number)

2× multiply two by that (gives 0, 2, 4, or 6)

'thstndrd'↓⍨drop that many characters from this string

2↑ take the first two of the remaining characters

⍕, concatenate the stringified argument to that

APL (Dyalog Unicode), 38 36 bytes

Thanks to ngn for fixing a bug while maintaining byte count.

Anonymous tacit prefix function. Requires ⎕IO (Index Origin) set to 0, which is default on many systems. Even works for 0!

⍕,{2↑'thstndrd'↓⍨2×⊃⍵⌽∊1 0 8\⊂10↑⍳4}

Try it online!

{…} anonymous lambda; ⍵ is argument:

⍳4 first four ɩndices; [0,1,2,3]

10↑ take first ten elements from that, padding with zeros: [0,1,2,3,0,0,0,0,0,0]

⊂ enclose to treat as single element; [[0,1,2,3,0,0,0,0,0,0]]

1 0 8\ expand to one copy, a prototypical copy (all-zero), eight copies;
[[0,1,2,3,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0],
[0,1,2,3,0,0,0,0,0,0],
[0,1,2,3,0,0,0,0,0,0],
⋮ (5 more)
[0,1,2,3,0,0,0,0,0,0]]

∊ ϵnlist (flatten);
[0,1,2,3,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,
0,1,2,3,0,0,0,0,0,0,
0,1,2,3,0,0,0,0,0,0,
⋮ (50 more)
0,1,2,3,0,0,0,0,0,0]

⍵⌽ cyclically rotate left as many steps as indicated by the argument

⊃ pick the first number (i.e. the argument-mod-100'th number)

2× multiply two by that (gives 0, 2, 4, or 6)

'thstndrd'↓⍨drop that many characters from this string

2↑ take the first two of the remaining characters

⍕, concatenate the stringified argument to that

#Dyalog APL, 38 36 bytes

⍕,{⊃⌽↓⍵2⍴('stndrd'/⍨9≥20|⍵),14⍴'th'}

Works by generating two-letter strings for all numbers up to the given one, then returns the last string. If the given number is 9 or up, it does not generating any st/nd/rd's at all, so 11th/12th/13th become right.

This whole function is one single so-called fork: it applies the ⍕ function (which makes a number into a string) to the argument, and applies the anonymous {…} function (explained below) to the argument, and then it concatenates (,) the two results.

Explanation for the {…} function:

14⍴'th' recycle the letters until a string of length 14 is reached
(…), prepends the parenthesized value
20|⍵ the remainder when divided by 20
9≥ boolean if 9 is greater than or equal to
'stndrd'/⍨ the string if true; an empty string if false
⍵2⍴ makes into an argument-by-2 table
↓ transforms the table into a list of strings
⊃⌽ last sting (literally the first of the reversed list)

APL (Dyalog Unicode), 38 36 bytes

Thanks to ngn for fixing a bug while maintaining byte count.

Anonymous tacit prefix function. Requires ⎕IO (Index Origin) set to 0, which is default on many systems.

⍕,{2↑'thstndrd'↓⍨2×⊃⍵⌽∊1 0 8\⊂10↑⍳4}

Try it online!

{…} anonymous lambda; ⍵ is argument:

⍳4 first four ɩndices; [0,1,2,3]

10↑ take first ten elements from that, padding with zeros: [0,1,2,3,0,0,0,0,0,0]

⊂ enclose to treat as single element; [[0,1,2,3,0,0,0,0,0,0]]

1 0 8\ expand to one copy, a prototypical copy (all-zero), eight copies;
[[1,2,3,4,0,0,0,0,0,0], 
[0,0,0,0,0,0,0,0,0,0], 
[1,2,3,4,0,0,0,0,0,0], 
[1,2,3,4,0,0,0,0,0,0], 
⋮ (5 more)
[1,2,3,4,0,0,0,0,0,0]]

∊ ϵnlist (flatten);
[1,2,3,4,0,0,0,0,0,0, 
0,0,0,0,0,0,0,0,0,0, 
1,2,3,4,0,0,0,0,0,0,
1,2,3,4,0,0,0,0,0,0,
⋮ (50 more)
1,2,3,4,0,0,0,0,0,0]

⍵⌽ cyclically rotate left as many steps as indicated by the argument

⊃ pick the first number (i.e. the argument-mod-100'th number)

2× multiply two by that (gives 0, 2, 4, or 6)

'thstndrd'↓⍨drop that many characters from this string

2↑ take the first two of the remaining characters

⍕, concatenate the stringified argument to that

Dyalog APL, 38 36 bytes

⍕,{⊃⌽↓⍵2⍴('stndrd'/⍨9≥20|⍵),14⍴'th'}

Works by generating two-letter strings for all numbers up to the given one, then returns the last string. If the given number is 9 or up, it does not generating any st/nd/rd's at all, so 11th/12th/13th become right.

This whole function is one single so-called fork: it applies the ⍕ function (which makes a number into a string) to the argument, and applies the anonymous {…} function (explained below) to the argument, and then it concatenates (,) the two results.

Explanation for the {…} function:

14⍴'th' recycle the letters until a string of length 14 is reached
(…), prepends the parenthesized value
20|⍵ the remainder when divided by 20
9≥ boolean if 9 is greater than or equal to
'stndrd'/⍨ the string if true; an empty string if false
⍵2⍴ makes into an argument-by-2 table
↓ transforms the table into a list of strings
⊃⌽ last sting (literally the first of the reversed list)

#Dyalog APL, 38 36 bytes

⍕,{⊃⌽↓⍵2⍴('stndrd'/⍨9≥20|⍵),14⍴'th'}

Works by generating two-letter strings for all numbers up to the given one, then returns the last string. If the given number is 9 or up, it does not generating any st/nd/rd's at all, so 11th/12th/13th become right.

This whole function is one single so-called fork: it applies the ⍕ function (which makes a number into a string) to the argument, and applies the anonymous {…} function (explained below) to the argument, and then it concatenates (,) the two results.

Explanation for the {…} function:

14⍴'th' recycle the letters until a string of length 14 is reached
(…), prepends the parenthesized value
20|⍵ the remainder when divided by 20
9≥ boolean if 9 is greater than or equal to
'stndrd'/⍨ the string if true; an empty string if false
⍵2⍴ makes into an argument-by-2 table
↓ transforms the table into a list of strings
⊃⌽ last sting (literally the first of the reversed list)