Ramon Navaratnam, the chairman of the Kuala Lumpur-based Centre for Public Policy Studies, said the polls were crucial because they were being seen as "a proxy for the popularity, confidence and faith in the future leadership of Najib".
Najib has pledged far-reaching administrative and social reforms to revive the government's support following the ruling Barisan Nasional's (BN) worst general elections in four decades last year.
But with a resurgent opposition, an economy on the brink of recession, and political, racial and social tensions at their highest in years, the new prime minister has his work cut out.
Still, the opposition is "not as confident" as it was last year when it made strong gains in the general election, said Ooi Kee Beng, a political analyst at the Institute of Southeast Asian Studies in Singapore.
The seat in Perak will be watched particularly closely after the BN wrested back the state from the opposition in January following three opposition legislators crossing over.
And while the opposition cannot retake the state even if it wins Tuesday's by-election there, a loss for Najib's coalition could signal voters' rejection of the new leader and the BN's method of retaking the state.