The easiest way to look "under the hood" of AES is likely one of the many AES in Excel spreadsheets that are floating around: https://www.nayuki.io/page/aes-cipher-internals-in-excel
You will also need to sort out how the initial message was padded because the AES block size is 128-bits, and "Hello" is not 128-bits. There is a value to going through the s-box mathematics and the flow, but I would not suggest that anyone do the cipher completely by hand.
As others have said, yes, you can do it by hand. I'm not aware of any task a conventional (ie not quantic) computer can do and you cannot. However, it could be that some tasks they can perform in one second would take a human an entire life.
Also, as pointed in one comment, what you have here is not "pure" AES. AES takes 128 bits blocks, and keys of varied size (64, 128, 256). Your input, assuming UTF-8 encoding, would be 32 bits long (and usually would be padded), your key assuming UTF-8 is also 32 bits long (could also have been using as 16 bits integer). And your output is 32 "characters", which assuming UTF-8 is 256 bits. Your output, very surprisingly, is also entirely made of "printable characters" (letters, symbols and digits). It is very unlikely that AES would have such an output (except maybe some of the bad modes that are to be avoided). I thus highly suspect your output is encoded using... something... It "looks like" Base64 (it is compatible with its default charset), except it's too long to encode 128 bits.
Conclusion: yes it can be done, but don't try comparing what you'd do by hand with what you have here. Here, some extra processing has been done.
It would be very long (if it is even feasible) given that it continues at least ten rounds.
If you want to do something manageable, you can practice with the simplified aes, which is made precisely for practicing by hand tasks. It continues only two rounds and reduced parameters.
If you want to decode the real AES, my advice is to try to do one round of it and see if it is manageable with its real parameters.