Measurements of multiplicity and transverse momentum fluctuations of charged particles were performed in inelastic p+p interactions at 20, 31, 40, 80, and 158 GeV /c beam momentum. Results for the scaled variance of the multiplicity distribution and for three strongly intensive measures of multiplicity and transverse momentum fluctuations Δ[P T ,N], Σ[P T ,N] and Φ p T are presented. For the first time the results on fluctuations are fully corrected for experimental biases. The results on multiplicity and transverse momentum fluctuations significantly deviate from expectations for the independent particle production. They also depend on charges of selected hadrons. The string-resonance Monte Carlo models Epos and Urqmd do not describe the data. The scaled variance of multiplicity fluctuations is significantly higher in inelastic p+p interactions than in central Pb+Pb collisions measured by NA49 at the same energy per nucleon. This is in qualitative disagreement with the predictions of the Wounded Nucleon Model. Within the statistical framework the enhanced multiplicity fluctuations in inelastic p+p interactions can be interpreted as due to event-by-event fluctuations of the fireball energy and/or volume.