#Javascript (ES6), 80 91 90 bytes

a=>[...a].reduce((p,c)=>c=='!'?p^255:c=='<'?p*2%256:c=='>'?p>>1:c=='@'?p/16|0+p%16*16:p,0)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ~ would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, truly shifts the bits of x 1 bit to the right.
• For @, floors x/16 and adds it to x%16*16.

Thanks to @vihan for suggesting the use of reduce to save a byte.

Javascript (ES6), 80 91 90 bytes

a=>[...a].reduce((p,c)=>c=='!'?p^255:c=='<'?p*2%256:c=='>'?p>>1:c=='@'?p/16|0+p%16*16:p,0)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ~ would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, truly shifts the bits of x 1 bit to the right.
• For @, floors x/16 and adds it to x%16*16.

Thanks to @vihan for suggesting the use of reduce to save a byte.

#Javascript (ES6), 80 91 bytes

a=>(x=0,[...a].map(c=>x=c=='!'?x^255:c=='<'?x*2%256:c=='>'?x>>1:c=='@'?x/16|0+x%16*16:x),x)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ~ would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, truly shifts the bits of x 1 bit to the right.
• For @, floors x/16 and adds it to x%16*16.

#Javascript (ES6), 80 91 90 bytes

a=>[...a].reduce((p,c)=>c=='!'?p^255:c=='<'?p*2%256:c=='>'?p>>1:c=='@'?p/16|0+p%16*16:p,0)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ~ would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, truly shifts the bits of x 1 bit to the right.
• For @, floors x/16 and adds it to x%16*16.

Thanks to @vihan for suggesting the use of reduce to save a byte.

#Javascript (ES6), 80 92 bytes

a=>(x=0,[...a].map(c=>x=c=='!'?x^255:c=='<'?x*2%256:c=='>'?x>>1:c=='@'?x/16|0+x%16*16:x),x)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ! would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, divides x by 2 and floors the result.
• For @, floors x/16 and adds it to x%16*16.

#Javascript (ES6), 80 91 bytes

a=>(x=0,[...a].map(c=>x=c=='!'?x^255:c=='<'?x*2%256:c=='>'?x>>1:c=='@'?x/16|0+x%16*16:x),x)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ~ would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, truly shifts the bits of x 1 bit to the right.
• For @, floors x/16 and adds it to x%16*16.