Can the president spend money without Congress' approval?
The President cannot spend money without the approval of Congress and cannot directly authorize the spending of money.
The president makes an annual budget request to congress describing how the various departments of the executive branch will use the budget. There is also discretionary spending which has to be asked for annually which and the president irons out with congressional Appropriations Committees. Extra defense spending, for example, has to be asked for individually. The various departments that fall under POTUS recommendations include Education, Housing, Treasury, State (Foreign), Interior, etc.
Congress brings in Dept Secretaries and questions them and it goes in/out of committee before it goes to the floor of the two houses. It passes by simple majority (after pork, or, paper-clipped add-ons mysteriously appear, usually to benfit one congressman's region). Because it is a resolution, not a bill, it does not go back to POTUS to be signed/vetoed. It just goes into effect from congress.
From the dot gov:
" the President's budget must request a specific funding level for appropriated programs and may also request changes in tax and entitlement law."
The entitlement programs are:
Social Security, Medicare, Medicaid, Nutrition Assistance, Retirement Benefits, Veterans' Disability, and Unemployment Insurance
This is an oversimplification that doesn't take into account, for example, Reconciliation. The entire process, enveloping all three branches of government is difficult even for congressman to learn fully. The president has a team of advisers just for budgetary spending. Congress has the Congressional Budget Office, committee experts, advisers, and aides. It's all part of the "Balance of Power" and the current budget procedures date to the Nixon era (except for the Byrd rule, which is also more complicated).