It's a yearly task, alright

PHP, 38 40 30 28 bytes

<?=date("jS F",86399*$argn);

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Run with php -nF input is from STDIN. Example (above script named y.php):

$ echo 1|php -nF y.php
1st January
$ echo 2| php -nF y.php
2nd January
$ echo 3| php -nF y.php
3rd January
$ echo 11|php -nF y.php
11th January
$ echo 21|php -nF y.php
21st January
$ echo 60|php -nF y.php
1st March
$ echo 365|php -nF y.php
31st December

Explanation

Construct an epoch timestamp for the desired day in 1970 (conveniently not a leap year) by multiplying the day number * number of seconds per day (86400). However, this would yield one day higher so instead multiply by number of seconds in a day - 1 (86399) which for the range of input numbers (1≤n≤365) will result with the timestamp of the end of each correct day. Then just use PHP's built-in date formatting for output.

C# (Visual C# Interactive Compiler), 115 113 109 98 bytes

g=>$"{f=(g=p.AddDays(g-1)).Day}{"tsnr"[f=f%30%20<4?f%10:0]}{"htdd"[f]} {g:MMMM}";DateTime p;int f;

Thanks to @someone for saving 9 bytes

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Jelly,  79 78  77 bytes

-1 fixing a bug :) _{(shouldn't pre-transpose to find index, should post-reverse, but then we can tail rather than head)}
-1 using reflection _{(⁽©ṅB+30_2¦2 -> ⁽0ṗb4+28m0)}

⁽0ṗb4+28m0SRṁRƲœiµṪȮ%30%20«4ị“nḄƲf⁷»s3¤Ṗ,ị“£ṢtẒ⁽ẹ½MḊxɲȧėAṅ ɓaṾ¥D¹ṀẏD8÷ṬØ»Ḳ¤$K

A full program which prints the result

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How?

will update this later...

⁽©ṅB+30_2¦2SRṁRƲZœiµḢȮ%30%20«4ị“nḄƲf⁷»s3¤Ṗ,ị“...»Ḳ¤$K - Main Link: integer, n
⁽©ṅB+30_2¦2SRṁRƲZœi - f(n) to get list of integers, [day, month]
⁽©ṅ                 - compressed literal 2741
   B                - to a list of binary digits -> [ 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1]
    +30             - add thirty                    [31,30,31,30,31,30,31,31,30,31,30,31]
         ¦          - sparse application...
        2           - ...to indices: [2]
       _  2         - ...action: subtract two       [31,28,31,30,31,30,31,31,30,31,30,31]
               Ʋ    - last four links as a monad - i.e. f(x):
           S        -   sum x                       365
            R       -   range                       [1..365]
              R     -   range x (vectorises)        [[1..31],[1..28],...]
             ṁ      -   mould like                  [[1..31],[32..59],...]
                Z   - transpose                     [[1,32,...],[2,33,...],...]
                 œi - 1st multi-dimensional index of n  -> [day, month]

µḢȮ%30%20«4ị“nḄƲf⁷»s3¤Ṗ,ị“...»Ḳ¤$K - given [day, month] format and print
µ                                  - start a new monadic chain - i.e. f(x=[day, month])
 Ḣ                                 - head -- get the day leaving x as [month])
  Ȯ                                - print it (with no newline) and yield it
   %30                             - modulo by thirty
      %20                          - modulo by twenty
         «4                        - minimum of that and four
                     ¤             - nilad followed by link(s) as a nilad:
            “nḄƲf⁷»                -   dictionary words "standard"+" the" = "standard the"
                   s3              -   split into threes = ["sta","nda","rd ","the"]
           ị                       - index into
                      Ṗ            - remove rightmost character
                               ¤   - nilad followed by link(s) as a nilad:
                         “...»     -   dictionary words "January"+" February"+...
                              Ḳ    -   split at spaces = ["January","February",...]
                        ị          - index into (vectorises across [month])
                       ,           - pair                  e.g. ["th", ["February"]]
                                K  - join with spaces           ["th ", "February"]
                                   - print (implicitly smashes)   th February