Retina, 46 bytes

/[1-3]{2}|./_T`d`〇〡-〩`^.

T`123`一二三

Try it online! Link includes test cases. Explanation:

/[1-3]{2}|./

Match either two digits 1-3 or any other digit.

_T`d`〇〡-〩`^.

Replace the first character of each match with its Suzhou.

T`123`一二三

Replace any remaining digits with horizontal Suzhou.

51 bytes in Retina 0.8.2:

M!`[1-3]{2}|.

mT`d`〇〡-〩`^.

T`¶123`_一二三

Try it online! Link includes test cases. Explanation:

M!`[1-3]{2}|.

Split the input into individual digits or pairs of digits if they are both 1-3.

mT`d`〇〡-〩`^.

Replace the first character of each line with its Suzhou.

T`¶123`_一二三

Join the lines back together and replace any remaining digits with horizontal Suzhou.

JavaScript, 81 bytes

s=>s.replace(/./g,c=>(p=14>>c&!p)|c>3?eval(`"\u302${c}"`):'〇一二三'[c],p=0)

Try it online!

Using 14>>c saves 3 bytes. Thanks to Arnauld.

JavaScript (ES6),  95 89  88 bytes

Saved 6 bytes thanks to @ShieruAsakoto

Takes input as a string.

s=>s.replace(i=/./g,c=>'三二一〇〡〢〣〤〥〦〧〨〩'[i=112>>i&c<4?3-c:+c+3])

Try it online!

Clean, 181 165 bytes

All octal escapes can be replaced by the equivalent single-byte characters (and are counted as one byte each), but used for readability and because otherwise it breaks TIO and SE with invalid UTF-8.

import StdEnv

u=mapc={'343','200',c}

?s=((!!)["〇":s++u['244245246247250']])o digitToInt

$=

$[h:t]=[?(u['241242243'])h:if(h-'1'<'03')f$t]

f=

f[h:t]=[?["一","二","三"]h: $t]

Try it online!

An encoding-unaware compiler is both a blessing and a curse.

Java (JDK), 120 bytes

s->{for(int i=0,p=0,c;i<s.length;)s[i]+=(p>0&p<4&(c=s[i++]-48)>0&c<4)?"A䷏乚䷖".charAt(c+(p=0)):(p=c)<1?12247:12272;}

Try it online!

Credits

• -3 bytes thanks to Kevin Cruijssen
  

Perl 5 -pl -Mutf8, 53 46 bytes

-7 bytes thanks to Grimy

s/[123]{2}|./OS&$&/ge;y//〇〡-〰一二三/c

Try it online!

Explanation

# Binary AND two consecutive digits 1-3 (ASCII 0x31-0x33)

# or any other single digit (ASCII 0x30-0x39) with string "OS"

# (ASCII 0x4F 0x53). This converts the first digit to 0x00-0x09

# and the second digit, if present, to 0x11-0x13.

s/[123]{2}|./OS&$&/ge;

# Translate empty complemented searchlist (0x00-0x13) to

# respective Unicode characters.

y//〇〡-〰一二三/c

Perl 6 -p, 85 61 bytes

-13 bytes thanks to Jo King

s:g[(1|2|3)<((1|2|3)]=chr $/+57;tr/0..</〇〡..〩一二三/

Try it online!

Red, 198 171 bytes

func[n][s: charset"〡〢〣"forall n[n/1: either n/1 >#"0"[to-char 12272 + n/1][#"〇"]]parse

n[any[[s change copy t s(pick"一二三"do(to-char t)- 12320)fail]| skip]]n]

Try it online!

R, 138 bytes

I'll bet there's an easier way to do this. Use gsub to get the alternating numeric positions.

function(x,r=-48+~x)Reduce(paste0,ifelse(58<~gsub("[123]{2}","0a",x),"123"["一二三",r],'0-9'["〇〡-〩",r]))

"~"=utf8ToInt

"["=chartr

Try it online!

Python 3, 102 bytes

f=0

for i in input():f=i in'123'and 9-f;print(end='〇一二三〤〥〦〧〨〩〡〢〣'[int(i)+f])

Try it online!

mypetlion reminded me of a trivial golf. -4 bytes.

Ruby -p, 71 bytes

$_=gsub(/[1-3]K[1-3]/){|x|(x.ord+9).chr}.tr"0-<","〇〡-〩一二三"

Try it online!

K (ngn/k), 67 bytes

{,/(0N 3#"〇一二三〤〥〦〧〨〩〡〢〣")x+9*<x&x<4}@10

Try it online!

10 get list of decimal digits

{ }@ apply the following function

x&x<4 boolean (0/1) list of where the argument is less than 4 and non-zero

< scan with less-than. this turns runs of consecutive 1s into alternating 1s and 0s

x+9* multiply by 9 and add x

juxtaposition is indexing, so use this as indices in...

0N 3#"〇一二三〤〥〦〧〨〩〡〢〣" the given string, split into a list of 3-byte strings. k is not unicode aware, so it sees only bytes

,/ concatenate

C#, 127 bytes

n=>Regex.Replace(n+"",@"[4-90]|[1-3]{1,2}",x=>"〇〡〢〣〤〥〦〧〨〩"[x.Value[0]-'0']+""+(x.Value.Length>1?"一二三"[x.Value[1]-'0'-1]+"":""))

Try it online!

Jelly, 38 bytes

9Rż“øƓ“œ%“øƈ’;-25+⁽-EỌœị@DżD<4«Ɗ‘×¥ƊƊ

Try it online!

Jelly, 35 bytes

9Ḷ;-26ż“/Ẉ8‘+⁽ȷc¤ṃ@ɓD_2ỊŒgÄFị"+⁽-FỌ

Try it online!

Wolfram Language (Mathematica), 117 bytes

FromCharacterCode[12320+(IntegerDigits@#/. 0->-25//.MapIndexed[{a___,c=#2[[1]],c,b___}->{a,c,#,b}&,{0,140,9}+7648])]&

Try it online!

Note that on TIO this outputs the result in escaped form. In the normal Wolfram front end, it will look like this: