This is rather meant as a demonstration, not a full-featured syntax file. But it should work correctly against an example above.

~/.vim/syntax/objdump.vim

" standard guard against syntax clash
if exists("b:current_syntax")
    finish
endif

" Asm is our main syntax; C is an extra one
runtime! syntax/asm.vim
unlet b:current_syntax
syntax include @C syntax/c.vim
unlet b:current_syntax

" a line starting with [A-Za-z_] is a C line
syntax region odCLine start=/^\h/ end=/$/ keepend contains=@C

" a line starting with a hex number is an objdump address/opcode
" stop at [:xdigit:][:blank:][:blank:] to interpret the rest as asm instruction (default)
syntax region odHead start='^\s*\x\+\>' end=/\x\s\{2,}\|$/ keepend contains=odHex,asmIdentifier

" hex number w/o any prefix (deadbeef)
syntax match odHex '\<\x\+\>' contained
hi def link odHex hexNumber

" assign current syntax
let b:current_syntax = 'objdump'

Of course, you still need filetype/ftdetect to make it working.

This is rather meant as a demonstration, not a full-featured syntax file. But it should work correctly against an example above (except only few assemblers, such as GNU, support C/C++ comments natively; in general such comments should be marked as C syntax region).

~/.vim/syntax/objdump.vim

" standard guard against syntax clash
if exists("b:current_syntax")
    finish
endif

" Asm is our main syntax; C is an extra one
runtime! syntax/asm.vim
unlet b:current_syntax
syntax include @C syntax/c.vim
unlet b:current_syntax

" a line starting with [A-Za-z_] is a C line
syntax region odCLine start=/^\h/ end=/$/ keepend contains=@C

" a line starting with a hex number is an objdump address/opcode
" stop at [:xdigit:][:blank:][:blank:] to interpret the rest as asm instruction (default)
syntax region odHead start='^\s*\x\+\>' end=/\x\s\{2,}\|$/ keepend contains=odHex,asmIdentifier

" hex number w/o any prefix (deadbeef)
syntax match odHex '\<\x\+\>' contained
hi def link odHex hexNumber

" assign current syntax
let b:current_syntax = 'objdump'

Of course, you still need filetype/ftdetect to make it working.