Name the Aztec year

Python 3, ^{68 62} 59 bytes

Saved 6 9 bytes thanks to my pronoun is monicareinstate!!!

lambda n:((n+2)%13+1,"TCTAeaocclcaplhtaitlt l l i"[n%4::4])

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dc, 63 62 characters

[Tecpatl]0:y[Calli]1:y[Tochtli]2:y[Acatl]3:y?d2+13%1+n32P4%;yp

Explained:

[Tecpatl] 0 :y    # store the string "Tecpatl" in array y at index 0
[Calli]   1 :y    # store the string "Calli" in array y at index 1
[Tochtli] 2 :y    # store the string "Tochtli" in array y at index 2
[Acatl]   3 :y    # store the string "Acatl" in array y at index 3
?                 # read input input
d                 # duplicate it to keep a copy for later
2+ 13% 1+ n       # the usual formula: (year + 2) % 13 + 1, then print it
32 P              # print a space
4 %               # use the left copy of input for formula: year % 4
;y p              # get the value from array y at the given index, then print it

Sample run:

bash-5.0$ dc -e '[Tecpatl]0:y[Calli]1:y[Tochtli]2:y[Acatl]3:y?d2+13%1+n[ ]n4%;yp' <<< 1111
9 Acatl

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perl -pl, 61 bytes

$_=(($_+3)%13||13).$".(qw[Tecpatl Calli Tochtli Acatl])[$_%4]

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Nothing special going on, modding the year twice, with some aligning. $" is a variable which by default holds a space.