The president can veto a bill, and his veto can get vetoed by a popular enough vote.
Sure. Demand that your party not overturn your veto (since you have the Senate).
Obviously. However, he can veto spending cuts. Had Obama done so (and provided aid to the States so they didn't need to lay off teachers and so on) the unemployment rate would be around 5%.
Or it could be much worse as the GOP could have hardened their position and we could have had one of more government shut downs. Even without that there was no way Obama was going to get any new spending had he vetoed every cut.
There isn't really any need for new spending at this point. As Obama has himself been misquoted pointing out, the private sector is basically recovered. The only jobs missing from the economy are state jobs, largely state school teachers*, but also all those guys laid off through the various sequester too-bad-too-happen catastrophes that have been inflicted on you guys over the last couple of years.
Obviously this shit is mostly the Republicans' fault, no question. The outrageous irresponsibility of threatening anarchy unless you get your way is difficult to deal with. But, I'm sure, Obama probably could have stopped this if he'd kept his head: look what the Republicans are doing! This is totally irresponsible! Now he's getting blamed for it.
* This is also Obama's fault since when these cuts took place he had both House and unfillibusterable Senate. 500 billion or a trillion in aid to states and this wouldn't have happened.