Between 1955-57, the POTUS was Mr. Dwight Eisenhower who belonged to the Republican Party. But, at the same time, the majority of both the senate and House of Rep. belonged to the Democrats. How difficult was it to sign a bill for the President when both the houses will vote against the president? Does he need to invoke executive power to pass any bill?
You seem to misunderstand the process. Bills originate in congress, and the president serves as a gatekeeper deciding whether to accept bills passed by congress. The president can ask congress to pass a certain provision, but congress is under no obligation to accommodate such requests.
So first, one house passes a bill which has been introduced by one of its members (and probably has been approved by one or more committees before it can be brought to a vote). Second, the other house does the same, also subject to the committee process. Only after both houses of congress have passed a bill is it sent to the president for consideration. The president may sign the bill or reject it.
In the latter case, where there is a disagreement between congress and the president, only congress can overcome the disagreement, by overriding the veto, which requires each house to vote by a supermajority of two thirds or more.
The past is a different country...
Back in the 50s the country was much more bipartisan, with support for the President on many issues coming from both Republicans and Democrats. Both parties were quite different from what they are now. Both had liberal and conservative wings and what it meant to be "liberal" or "conservative" was quite different from what it means now. The racial aspect of American politics was also quite different, with openly racist politicians in both parties, but particularly among the Democrats.
So, the president needed to negotiate with the Democrats, there was horse-trading and quid-pro-quo. But it simply wasn't the case that "both houses would vote against the President".
There is no executive power to pass a bill, the executive only has the power to block (veto) a bill and ask the Legislature to reconsider.