TypeScript's type system, 206 bytes

//@ts-ignore
type F<C,S=[0,0]>=[S,C]extends[[infer A,infer B,...infer V],[infer I,...infer R]]?F<R,I extends`${any}`?[I,...S]:[...[[B],[A,A,B],[B,A],[`${A}${B}`],B extends`${A}${infer Z}`?[Z]:0][I],...V]>:S

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TypeScript's type system, 206 205 bytes

//@ts-ignore
type F<C,S=[0,0]>=[S,C]extends[[infer A,infer B,...infer V],[infer I,...infer R]]?F<R,I extends string?[I,...S]:[...[[B],[A,A,B],[B,A],[`${A}${B}`],B extends`${A}${infer Z}`?[Z]:0][I],...V]>:S

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type F<
  // C is the input string type
  C,
  // S is the stack, starts with two useless elements
  S=[0,0],
>=[S,C]extends[
    // get the top 2, A and B, and the rest V, of the stack
    [infer A,infer B,...infer V],
    // get the first word I and the rest R of C
    `${infer I} ${infer R}`
  ]
    // recurse, setting C to R, and setting S to...
    ?F<R,
      // does I start with a 1?
      I extends`1${any}`
        // if so, prepend it to S
        ?[I,...S]
        // otherwise, it's a command; index into this object
        // to replace the top 2 of the stack:
        :[...{
          // pop- just the second
          p:[B],
          // dup- first twice, then second
          d:[A,A,B],
          // swap: second, then first
          x:[B,A],
          // add: concatenate the first and second
          a:[`${A}${B}`],
          // subtract: the second with the first taken away
          s:B extends`${A}${infer Z}`?Z:0
        // then append V
        }[I],...V]
    >
  // if C is empty (i.e. program finished), return S
  :S

type F<
  // C is the input string type
  C,
  // S is the stack, starts with two useless elements
  S=[0,0],
>=[S,C]extends[
    // get the top 2, A and B, and the rest V, of the stack
    [infer A,infer B,...infer V],
    // get the first word I and the rest R of C
    `${infer I} ${infer R}`
  ]
    // recurse, setting C to R, and setting S to...
    ?F<R,
      // does I start with a 1?
      I extends`1${any}`
        // if so, prepend it to S
        ?[I,...S]
        // otherwise, it's a command; index into this object
        // to replace the top 2 of the stack:
        :[...{
          // pop- just the second
          p:[B],
          // dup- first twice, then second
          d:[A,A,B],
          // swap: second, then first
          x:[B,A],
          // add: concatenate the first and second
          a:[`${A}${B}`],
          // subtract: the second with the first taken away
          s:B extends`${A}${infer Z}`?[Z]:0
        // then append V
        }[I],...V]
    >
  // if C is empty (no more commands are left), return S
  :S

TypeScript’s type system, 324 309 251 220 bytes

This is a generic type F<S> which takes input as a string type of space separated commands or unary numbers, space terminated. Output in unary, first element is top of stack.

TypeScript's type system, 206 bytes

//@ts-ignore
type F<C,S=[0,0]>=[S,C]extends[[infer A,infer B,...infer V],[infer I,...infer R]]?F<R,I extends`${any}`?[I,...S]:[...[[B],[A,A,B],[B,A],[`${A}${B}`],B extends`${A}${infer Z}`?[Z]:0][I],...V]>:S

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This is a generic type F<C> which takes input as a tuple of items which are either unary numbers as strings of 1s, or numbers 0 to 4 for the 5 commands:

This solution works identically to the one below, only it skips the step of splitting on spaces, and indexes into an array instead of a dictionary, so it ends up quite a bit shorter.

TypeScript’s type system, 324 309 251 220 bytes

This is a generic type F<C> which takes input as a string type of space separated commands or unary numbers, space terminated. Output in unary, first element is top of stack.