If the format of the string is fixed, and you always have just 3 [...] groups inside to deal with, you may define a block that matches [name:...] and captures the 2 parts into separate groups and use a quite simple code with .replaceAll:
String s = "City: [name:NYK][distance:1100] [name:CLT][distance:2300] [name:KTY][distance:3540] Price:";
String matchingBlock = "\\s*\\[name:([A-Z]+)]\\[distance:(\\d+)]";
String res = s.replaceAll(String.format(".*%1$s%1$s%1$s.*", matchingBlock),
"[$1:$2][$3:$4][$5:$6]");
System.out.println(res); // [NYK:1100][CLT:2300][KTY:3540]
See the Java demo and a regex demo.
The block pattern matches:
\\s* - 0+ whitespaces
\\[name: - a literal [name: substring
([A-Z]+) - Group n capturing 1 or more uppercase ASCII chars (\\w+ can also be used)
]\\[distance: - a literal ][distance: substring
(\\d+) - Group m capturing 1 or more digits
] - a ] symbol.
In the .*%1$s%1$s%1$s.* pattern, the groups will have 1 to 6 IDs (referred to with $1 - $6 backreferences from the replacement pattern) and the leading and final .* will remove start and end of the string (add (?s) at the start of the pattern if the string can contain line breaks).