If you want come from New Delhi to Patna, without any reservation then you can travel by link train without any reservation.
Here is best trains detail without any reservation.
(1) (A) Get train from New Delhi railway station to Kanpur Central, Train No. 12004 LJN SWRAN SHATABDI at 06.10 am (reaches Kanpur at 11.20 am). (B) Get train from Kanpur Central to DDU Junction, train number 18310 Jammu Tawi Sambalpur Express at 12.45 pm, (reaches Deen Dayal Upadhyaay Junction at 17.50 pm). (C) Get train from DDU Junction to Patna Junction, Train 13006 Amrtisar Howrah Mail at 18.13 pm, (Reaches Patna Junction at 21.30 pm same day). This is the shortest time taking train from New Delhi to Patna in one day.
(2) (A) Get train from Anand Vihar termianl (ANVT) to Lucknow Train number 12004 LJN SWRAN SHATABDI at 06.10 am (Reaches at Lucknow at 12.40 pm, (B) Get train from Lucknow to Varanasi, train number 14220 Lucknow Varanasi Intercity Express at 12.55 pm (Reaches Varanasi at 18.45 PM) (C) Get train from Varanasi, train number 13050 Aritsar Howrha Mail at 19.15 pm (Reaches Patna next day at 01.40 am).
(3) (A) Get Train from ANVT to DDU Junction, train number 12816 Nandan Kanan Express (Monday, Wednesday, Thursday and Saturday) at 06.30 am. (Reaches DDU junction at 18.00 pm.) (B) Get train from DDU Junction to Patna Junction, train number 13006 Amritsar Howrah Mail at 18.13 pm. (reaches Patna at 21.30 pm), or train number 12334 Vibhuti Express at 19.08 pm (reaches Patna at 22.35 pm on same day)————————————————————
(4) (A) Get Train from ANVT to DDU Junction, train number 12876 Neelanchal Express (Sunday, Tuesday & Friday) at 06.30 am. (Reaches DDU junction at 21.00 pm.) (B) Get train from DDU Junction to Patna Junction, train number 14224 BUDHPURNIMA EXP at 21.30 pm. (reaches Patna at 03.00 am).
(5) (A) Get train from Delhi railway station for Patna, train number 15484 Sikkim Mahananda Exp at 06.35 am (Reaches Patna at next day at 00.20 am)
(6) (A) Get train from NDLS to Sonepur, train number 12554 VAISHALI EXP at 19.50 pm (Reaches Sonepur at next day 13.30 pm). (B) Get bus from Sonepur to Patna which is 23 Km far from Sonepur.————————————————————
All above trains have general coaches, so you no need to get reservation in any trains.