The term "appropriations" may be the source of confusion here. In the US government, appropriations is the term they use for allocating budgeted money to specific spending. So the overall budget may allow for $500 billion in defense spending. The defense appropriations bill would say things like $50 billion for new bombers; $5 billion to maintain the existing nuclear missiles; $40 billion for submarines; etc. Note that appropriations bills will usually go into more detail than that.
An appropriations bill is exactly the kind of thing that can originate in the Senate.
Note that there is a separate issue that the Senate regularly ignores the rule against originating a tax bill by replacing an existing bill that modifies taxes in its entirety. The standing ruling is that this is allowable under the constitutional phrase "concur with amendments". This has nothing to do with appropriations, which occur after the money has been budgeted (which may require tax changes) and before it is spent (generally by the executive branch).